Вопросы и задания, страница 101 - гдз по математике 5 класс учебник Бунимович, Дорофеев

Глава 6. Делимость чисел. 21. Простые числа - страница 101.

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App\Models\Task {#1030 // resources/views/models/task/default.blade.php
  #connection: "mysql"
  #table: "tasks"
  #primaryKey: "id"
  #keyType: "int"
  +incrementing: true
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  #perPage: 15
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  #attributes: array:24 [
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    "updated_at" => null
    "field_page_start" => "101"
    "field_page_end" => null
    "field_url" => "/5-klass/matematika/bunimovich-uchebnik/vop-21"
    "field_display_title" => "Вопросы и задания"
    "field_outside_task" => null
    "field_task_type" => Illuminate\Database\Eloquent\Collection {#1036
      #items: []
      #escapeWhenCastingToString: false
    }
    "field_metatags_title" => null
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      #items: []
      #escapeWhenCastingToString: false
    }
    "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1078
      #items: array:1 [
        0 => App\Models\Branch {#1035
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          #table: "branches"
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          #keyType: "int"
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          #attributes: array:24 [
            "id" => 534092
            "created_at" => "2026-04-10 13:58:26"
            "updated_at" => null
            "field_display_title" => "Делимость чисел"
            "field_branch_order" => "6"
            "field_url" => null
            "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1038
              #items: array:1 [
                0 => App\Models\Term {#1037
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                  #attributes: array:7 [
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                    "updated_at" => null
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                    "field_cases" => array:6 [
                      "field_accusative_case" => "Главу"
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                      "field_genitive_case" => "Главы"
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            "field_page_start" => "95"
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                          #attributes: array:10 [ …10]
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                        }
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                    "field_class" => Illuminate\Database\Eloquent\Collection {#1043
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                        }
                      ]
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                    "field_publisher" => Illuminate\Database\Eloquent\Collection {#1045
                      #items: array:1 [
                        0 => App\Models\Term {#1046
                          #connection: "mysql"
                          #table: "terms"
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                          #attributes: array:6 [ …6]
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                          #changes: []
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                          #guarded: array:1 [ …1]
                        }
                      ]
                      #escapeWhenCastingToString: false
                    }
                    "field_author" => Illuminate\Database\Eloquent\Collection {#1047
                      #items: array:6 [
                        0 => App\Models\Term {#1048
                          #connection: "mysql"
                          #table: "terms"
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                          #keyType: "int"
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                        }
                        1 => App\Models\Term {#1049
                          #connection: "mysql"
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                          +timestamps: true
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                          #hidden: []
                          #visible: []
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                          #guarded: array:1 [ …1]
                        }
                        2 => App\Models\Term {#1050
                          #connection: "mysql"
                          #table: "terms"
                          #primaryKey: "id"
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                          #guarded: array:1 [ …1]
                        }
                        3 => App\Models\Term {#1051
                          #connection: "mysql"
                          #table: "terms"
                          #primaryKey: "id"
                          #keyType: "int"
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                          #guarded: array:1 [ …1]
                        }
                        4 => App\Models\Term {#1052
                          #connection: "mysql"
                          #table: "terms"
                          #primaryKey: "id"
                          #keyType: "int"
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                        }
                        5 => App\Models\Term {#1053
                          #connection: "mysql"
                          #table: "terms"
                          #primaryKey: "id"
                          #keyType: "int"
                          +incrementing: false
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                          #guarded: array:1 [ …1]
                        }
                      ]
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                    }
                    "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1054
                      #items: []
                      #escapeWhenCastingToString: false
                    }
                    "field_book_type" => Illuminate\Database\Eloquent\Collection {#1055
                      #items: array:1 [
                        0 => App\Models\Term {#1056
                          #connection: "mysql"
                          #table: "terms"
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                        }
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                    "field_country" => Illuminate\Database\Eloquent\Collection {#1057
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                          #table: "terms"
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                          #keyType: "int"
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                           …9
                        }
                      ]
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                    }
                    "field_city" => Illuminate\Database\Eloquent\Collection {#1059
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                        0 => App\Models\Term {#1060 …30}
                      ]
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                    }
                    "field_series" => Illuminate\Database\Eloquent\Collection {#1061
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                        0 => App\Models\Term {#1062 …30}
                      ]
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                    }
                    "field_umk" => Illuminate\Database\Eloquent\Collection {#1063
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                        0 => App\Models\Term {#1065 …30}
                      ]
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                    }
                    "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1066
                      #items: array:1 [
                        0 => App\Models\Term {#1067 …30}
                      ]
                      #escapeWhenCastingToString: false
                    }
                    "field_publication_number" => null
                    "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1068
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                    }
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                    "field_url" => "/5-klass/matematika/bunimovich-uchebnik"
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                    "breadcrumbs" => array:3 [
                      "class" => 376
                      "subject" => 544
                      "class_subject" => 398
                    ]
                  ]
                  #changes: []
                  #casts: []
                  #classCastCache: []
                  #attributeCastCache: []
                  #dateFormat: null
                  #appends: []
                  #dispatchesEvents: []
                  #observables: []
                  #relations: []
                  #touches: []
                  +timestamps: true
                  +usesUniqueIds: false
                  #hidden: []
                  #visible: []
                  #fillable: []
                  #guarded: array:1 [
                    0 => "*"
                  ]
                }
              ]
              #escapeWhenCastingToString: false
            }
            "branch_parent" => Illuminate\Database\Eloquent\Collection {#1075
              #items: []
              #escapeWhenCastingToString: false
            }
          ]
          #original: array:24 [
            "id" => 534092
            "created_at" => "2026-04-10 13:58:26"
            "updated_at" => null
            "field_display_title" => "Делимость чисел"
            "field_branch_order" => "6"
            "field_url" => null
            "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1038}
            "field_page_start" => "95"
            "field_branch_display" => "0"
            "field_branch_expanded" => "0"
            "field_display_branch_in_title" => "1"
            "field_display_task_interval" => "0"
            "field_display_branch_page" => "1"
            "field_branch_title_in_content" => "0"
            "field_navigation_title" => null
            "field_metatags_title" => null
            "field_metatags_description" => null
            "field_h1" => null
            "field_description_top" => null
            "field_description_bottom" => null
            "field_branch_cover" => []
            "field_branch_covers" => []
            "book" => Illuminate\Database\Eloquent\Collection {#1039}
            "branch_parent" => Illuminate\Database\Eloquent\Collection {#1075}
          ]
          #changes: []
          #casts: []
          #classCastCache: []
          #attributeCastCache: []
          #dateFormat: null
          #appends: []
          #dispatchesEvents: []
          #observables: []
          #relations: []
          #touches: []
          +timestamps: true
          +usesUniqueIds: false
          #hidden: []
          #visible: []
          #fillable: []
          #guarded: array:1 [
            0 => "*"
          ]
        }
      ]
      #escapeWhenCastingToString: false
    }
    "parent_branches" => Illuminate\Database\Eloquent\Collection {#1119
      #items: array:2 [
        0 => App\Models\Branch {#1127
          #connection: "mysql"
          #table: "branches"
          #primaryKey: "id"
          #keyType: "int"
          +incrementing: true
          #with: []
          #withCount: []
          +preventsLazyLoading: false
          #perPage: 15
          +exists: true
          +wasRecentlyCreated: false
          #escapeWhenCastingToString: false
          #attributes: array:24 [
            "id" => 534092
            "created_at" => "2026-04-10 13:58:26"
            "updated_at" => null
            "field_display_title" => "Делимость чисел"
            "field_branch_order" => "6"
            "field_url" => null
            "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1131
              #items: array:1 [
                0 => App\Models\Term {#1128
                  #connection: "mysql"
                  #table: "terms"
                  #primaryKey: "id"
                  #keyType: "int"
                  +incrementing: false
                  #with: []
                  #withCount: []
                  +preventsLazyLoading: false
                  #perPage: 15
                  +exists: true
                  +wasRecentlyCreated: false
                  #escapeWhenCastingToString: false
                  #attributes: array:7 [
                    "id" => 29
                    "created_at" => "2026-04-10 13:58:26"
                    "updated_at" => null
                    "name" => "Глава"
                    "field_cases" => array:6 [
                      "field_accusative_case" => "Главу"
                      "field_creative_case" => "Главой"
                      "field_dative_case" => "Главе"
                      "field_genitive_case" => "Главы"
                      "field_nominative_case" => "Глава"
                      "field_prepositional_case" => "Главе"
                    ]
                    "field_page_check_not_needed" => null
                    "field_short_name" => null
                  ]
                  #original: array:7 [
                    "id" => 29
                    "created_at" => "2026-04-10 13:58:26"
                    "updated_at" => null
                    "name" => "Глава"
                    "field_cases" => array:6 [
                      "field_accusative_case" => "Главу"
                      "field_creative_case" => "Главой"
                      "field_dative_case" => "Главе"
                      "field_genitive_case" => "Главы"
                      "field_nominative_case" => "Глава"
                      "field_prepositional_case" => "Главе"
                    ]
                    "field_page_check_not_needed" => null
                    "field_short_name" => null
                  ]
                  #changes: []
                  #casts: []
                  #classCastCache: []
                  #attributeCastCache: []
                  #dateFormat: null
                  #appends: []
                  #dispatchesEvents: []
                  #observables: []
                  #relations: []
                  #touches: []
                  +timestamps: true
                  +usesUniqueIds: false
                  #hidden: []
                  #visible: []
                  #fillable: []
                  #guarded: array:1 [
                    0 => "*"
                  ]
                }
              ]
              #escapeWhenCastingToString: false
            }
            "field_page_start" => "95"
            "field_branch_display" => "0"
            "field_branch_expanded" => "0"
            "field_display_branch_in_title" => "1"
            "field_display_task_interval" => "0"
            "field_display_branch_page" => "1"
            "field_branch_title_in_content" => "0"
            "field_navigation_title" => null
            "field_metatags_title" => null
            "field_metatags_description" => null
            "field_h1" => null
            "field_description_top" => null
            "field_description_bottom" => null
            "field_branch_cover" => []
            "field_branch_covers" => []
            "book" => Illuminate\Database\Eloquent\Collection {#1167
              #items: array:1 [
                0 => App\Models\Book {#1130
                  #connection: "mysql"
                  #table: "books"
                  #primaryKey: "id"
                  #keyType: "int"
                  +incrementing: false
                  #with: []
                  #withCount: []
                  +preventsLazyLoading: false
                  #perPage: 15
                  +exists: true
                  +wasRecentlyCreated: false
                  #escapeWhenCastingToString: false
                  #attributes: array:50 [
                    "id" => 59
                    "created_at" => "2026-04-10 13:58:26"
                    "updated_at" => null
                    "field_tree_default_status" => "tasks"
                    "field_pages_status" => null
                    "field_subject" => Illuminate\Database\Eloquent\Collection {#1132
                      #items: array:1 [
                        0 => App\Models\Term {#1129
                          #connection: "mysql"
                          #table: "terms"
                          #primaryKey: "id"
                          #keyType: "int"
                          +incrementing: false
                          #with: []
                          #withCount: []
                          +preventsLazyLoading: false
                          #perPage: 15
                          +exists: true
                          +wasRecentlyCreated: false
                          #escapeWhenCastingToString: false
                          #attributes: array:10 [
                            "id" => 6484
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Математика"
                            "field_abbreviated_name" => null
                            "field_cases" => array:6 [
                              "field_accusative_case" => "математику"
                              "field_creative_case" => "математикой"
                              "field_dative_case" => "математике"
                              "field_genitive_case" => "математики"
                              "field_nominative_case" => "математика"
                              "field_prepositional_case" => "математике"
                            ]
                            "field_foreign_lang_name" => null
                            "field_short_name" => null
                            "field_subject_type" => "technical_subject"
                            "field_translit" => "matematika"
                          ]
                          #original: array:10 [
                            "id" => 6484
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Математика"
                            "field_abbreviated_name" => null
                            "field_cases" => array:6 [
                              "field_accusative_case" => "математику"
                              "field_creative_case" => "математикой"
                              "field_dative_case" => "математике"
                              "field_genitive_case" => "математики"
                              "field_nominative_case" => "математика"
                              "field_prepositional_case" => "математике"
                            ]
                            "field_foreign_lang_name" => null
                            "field_short_name" => null
                            "field_subject_type" => "technical_subject"
                            "field_translit" => "matematika"
                          ]
                          #changes: []
                          #casts: []
                          #classCastCache: []
                          #attributeCastCache: []
                          #dateFormat: null
                          #appends: []
                          #dispatchesEvents: []
                          #observables: []
                          #relations: []
                          #touches: []
                          +timestamps: true
                          +usesUniqueIds: false
                          #hidden: []
                          #visible: []
                          #fillable: []
                          #guarded: array:1 [
                            0 => "*"
                          ]
                        }
                      ]
                      #escapeWhenCastingToString: false
                    }
                    "field_class" => Illuminate\Database\Eloquent\Collection {#1133
                      #items: array:1 [
                        0 => App\Models\Term {#1134
                          #connection: "mysql"
                          #table: "terms"
                          #primaryKey: "id"
                          #keyType: "int"
                          +incrementing: false
                          #with: []
                          #withCount: []
                          +preventsLazyLoading: false
                          #perPage: 15
                          +exists: true
                          +wasRecentlyCreated: false
                          #escapeWhenCastingToString: false
                          #attributes: array:6 [
                            "id" => 5445
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "5"
                            "field_cases" => array:6 [
                              "field_accusative_case" => "пятый"
                              "field_creative_case" => "пятым"
                              "field_dative_case" => "пятому"
                              "field_genitive_case" => "пятого"
                              "field_nominative_case" => "пятый"
                              "field_prepositional_case" => "пятом"
                            ]
                            "field_translit" => "pjatyj"
                          ]
                          #original: array:6 [
                            "id" => 5445
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "5"
                            "field_cases" => array:6 [
                              "field_accusative_case" => "пятый"
                              "field_creative_case" => "пятым"
                              "field_dative_case" => "пятому"
                              "field_genitive_case" => "пятого"
                              "field_nominative_case" => "пятый"
                              "field_prepositional_case" => "пятом"
                            ]
                            "field_translit" => "pjatyj"
                          ]
                          #changes: []
                          #casts: []
                          #classCastCache: []
                          #attributeCastCache: []
                          #dateFormat: null
                          #appends: []
                          #dispatchesEvents: []
                          #observables: []
                          #relations: []
                          #touches: []
                          +timestamps: true
                          +usesUniqueIds: false
                          #hidden: []
                          #visible: []
                          #fillable: []
                          #guarded: array:1 [
                            0 => "*"
                          ]
                        }
                      ]
                      #escapeWhenCastingToString: false
                    }
                    "field_publisher" => Illuminate\Database\Eloquent\Collection {#1135
                      #items: array:1 [
                        0 => App\Models\Term {#1136
                          #connection: "mysql"
                          #table: "terms"
                          #primaryKey: "id"
                          #keyType: "int"
                          +incrementing: false
                          #with: []
                          #withCount: []
                          +preventsLazyLoading: false
                          #perPage: 15
                          +exists: true
                          +wasRecentlyCreated: false
                          #escapeWhenCastingToString: false
                          #attributes: array:6 [
                            "id" => 5153
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Просвещение"
                            "field_cases" => null
                            "field_translit" => "prosveschenie"
                          ]
                          #original: array:6 [
                            "id" => 5153
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Просвещение"
                            "field_cases" => null
                            "field_translit" => "prosveschenie"
                          ]
                          #changes: []
                          #casts: []
                          #classCastCache: []
                          #attributeCastCache: []
                          #dateFormat: null
                          #appends: []
                          #dispatchesEvents: []
                          #observables: []
                          #relations: []
                          #touches: []
                          +timestamps: true
                          +usesUniqueIds: false
                          #hidden: []
                          #visible: []
                          #fillable: []
                          #guarded: array:1 [
                            0 => "*"
                          ]
                        }
                      ]
                      #escapeWhenCastingToString: false
                    }
                    "field_author" => Illuminate\Database\Eloquent\Collection {#1137
                      #items: array:6 [
                        0 => App\Models\Term {#1138
                          #connection: "mysql"
                          #table: "terms"
                          #primaryKey: "id"
                          #keyType: "int"
                          +incrementing: false
                          #with: []
                          #withCount: []
                          +preventsLazyLoading: false
                          #perPage: 15
                          +exists: true
                          +wasRecentlyCreated: false
                          #escapeWhenCastingToString: false
                          #attributes: array:12 [
                            "id" => 3941
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Бунимович"
                            "field_bio" => null
                            "field_degree_rp" => null
                            "field_foreign_lang_name" => null
                            "field_foreign_lang_patronymic" => null
                            "field_foreign_lang_surname" => null
                            "field_name" => "Евгений"
                            "field_patronymic" => "Абрамович"
                            "field_surname_rp" => null
                          ]
                          #original: array:12 [
                            "id" => 3941
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Бунимович"
                            "field_bio" => null
                            "field_degree_rp" => null
                            "field_foreign_lang_name" => null
                            "field_foreign_lang_patronymic" => null
                            "field_foreign_lang_surname" => null
                            "field_name" => "Евгений"
                            "field_patronymic" => "Абрамович"
                            "field_surname_rp" => null
                          ]
                          #changes: []
                          #casts: []
                          #classCastCache: []
                          #attributeCastCache: []
                          #dateFormat: null
                          #appends: []
                          #dispatchesEvents: []
                          #observables: []
                          #relations: []
                          #touches: []
                          +timestamps: true
                          +usesUniqueIds: false
                          #hidden: []
                          #visible: []
                          #fillable: []
                          #guarded: array:1 [
                            0 => "*"
                          ]
                        }
                        1 => App\Models\Term {#1139
                          #connection: "mysql"
                          #table: "terms"
                          #primaryKey: "id"
                          #keyType: "int"
                          +incrementing: false
                          #with: []
                          #withCount: []
                          +preventsLazyLoading: false
                          #perPage: 15
                          +exists: true
                          +wasRecentlyCreated: false
                          #escapeWhenCastingToString: false
                          #attributes: array:12 [
                            "id" => 4283
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Дорофеев"
                            "field_bio" => null
                            "field_degree_rp" => null
                            "field_foreign_lang_name" => null
                            "field_foreign_lang_patronymic" => null
                            "field_foreign_lang_surname" => null
                            "field_name" => "Георгий"
                            "field_patronymic" => "Владимирович"
                            "field_surname_rp" => null
                          ]
                          #original: array:12 [
                            "id" => 4283
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Дорофеев"
                            "field_bio" => null
                            "field_degree_rp" => null
                            "field_foreign_lang_name" => null
                            "field_foreign_lang_patronymic" => null
                            "field_foreign_lang_surname" => null
                            "field_name" => "Георгий"
                            "field_patronymic" => "Владимирович"
                            "field_surname_rp" => null
                          ]
                          #changes: []
                          #casts: []
                          #classCastCache: []
                          #attributeCastCache: []
                          #dateFormat: null
                          #appends: []
                          #dispatchesEvents: []
                          #observables: []
                          #relations: []
                          #touches: []
                          +timestamps: true
                          +usesUniqueIds: false
                          #hidden: []
                          #visible: []
                          #fillable: []
                          #guarded: array:1 [
                            0 => "*"
                          ]
                        }
                        2 => App\Models\Term {#1140
                          #connection: "mysql"
                          #table: "terms"
                          #primaryKey: "id"
                          #keyType: "int"
                          +incrementing: false
                          #with: []
                          #withCount: []
                          +preventsLazyLoading: false
                          #perPage: 15
                          +exists: true
                          +wasRecentlyCreated: false
                          #escapeWhenCastingToString: false
                          #attributes: array:12 [
                            "id" => 5879
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Суворова"
                            "field_bio" => null
                            "field_degree_rp" => null
                            "field_foreign_lang_name" => null
                            "field_foreign_lang_patronymic" => null
                            "field_foreign_lang_surname" => null
                            "field_name" => "Светлана"
                            "field_patronymic" => "Борисовна"
                            "field_surname_rp" => null
                          ]
                          #original: array:12 [
                            "id" => 5879
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Суворова"
                            "field_bio" => null
                            "field_degree_rp" => null
                            "field_foreign_lang_name" => null
                            "field_foreign_lang_patronymic" => null
                            "field_foreign_lang_surname" => null
                            "field_name" => "Светлана"
                            "field_patronymic" => "Борисовна"
                            "field_surname_rp" => null
                          ]
                          #changes: []
                          #casts: []
                          #classCastCache: []
                          #attributeCastCache: []
                          #dateFormat: null
                          #appends: []
                          #dispatchesEvents: []
                          #observables: []
                          #relations: []
                          #touches: []
                          +timestamps: true
                          +usesUniqueIds: false
                          #hidden: []
                          #visible: []
                          #fillable: []
                          #guarded: array:1 [
                            0 => "*"
                          ]
                        }
                        3 => App\Models\Term {#1141
                          #connection: "mysql"
                          #table: "terms"
                          #primaryKey: "id"
                          #keyType: "int"
                          +incrementing: false
                          #with: []
                          #withCount: []
                          +preventsLazyLoading: false
                          #perPage: 15
                          +exists: true
                          +wasRecentlyCreated: false
                          #escapeWhenCastingToString: false
                          #attributes: array:12 [
                            "id" => 4848
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Кузнецова"
                            "field_bio" => null
                            "field_degree_rp" => null
                            "field_foreign_lang_name" => null
                            "field_foreign_lang_patronymic" => null
                            "field_foreign_lang_surname" => null
                            "field_name" => "Людмила"
                            "field_patronymic" => "Викторовна"
                            "field_surname_rp" => null
                          ]
                          #original: array:12 [
                            "id" => 4848
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Кузнецова"
                            "field_bio" => null
                            "field_degree_rp" => null
                            "field_foreign_lang_name" => null
                            "field_foreign_lang_patronymic" => null
                            "field_foreign_lang_surname" => null
                            "field_name" => "Людмила"
                            "field_patronymic" => "Викторовна"
                            "field_surname_rp" => null
                          ]
                          #changes: []
                          #casts: []
                          #classCastCache: []
                          #attributeCastCache: []
                          #dateFormat: null
                          #appends: []
                          #dispatchesEvents: []
                          #observables: []
                          #relations: []
                          #touches: []
                          +timestamps: true
                          +usesUniqueIds: false
                          #hidden: []
                          #visible: []
                          #fillable: []
                          #guarded: array:1 [
                            0 => "*"
                          ]
                        }
                        4 => App\Models\Term {#1142
                          #connection: "mysql"
                          #table: "terms"
                          #primaryKey: "id"
                          #keyType: "int"
                          +incrementing: false
                          #with: []
                          #withCount: []
                          +preventsLazyLoading: false
                          #perPage: 15
                          +exists: true
                          +wasRecentlyCreated: false
                          #escapeWhenCastingToString: false
                          #attributes: array:12 [
                            "id" => 5198
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Минаева"
                            "field_bio" => null
                            "field_degree_rp" => null
                            "field_foreign_lang_name" => null
                            "field_foreign_lang_patronymic" => null
                            "field_foreign_lang_surname" => null
                            "field_name" => "Светлана"
                            "field_patronymic" => "Станиславовна"
                            "field_surname_rp" => null
                          ]
                          #original: array:12 [
                            "id" => 5198
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Минаева"
                            "field_bio" => null
                            "field_degree_rp" => null
                            "field_foreign_lang_name" => null
                            "field_foreign_lang_patronymic" => null
                            "field_foreign_lang_surname" => null
                            "field_name" => "Светлана"
                            "field_patronymic" => "Станиславовна"
                            "field_surname_rp" => null
                          ]
                          #changes: []
                          #casts: []
                          #classCastCache: []
                          #attributeCastCache: []
                          #dateFormat: null
                          #appends: []
                          #dispatchesEvents: []
                          #observables: []
                          #relations: []
                          #touches: []
                          +timestamps: true
                          +usesUniqueIds: false
                          #hidden: []
                          #visible: []
                          #fillable: []
                          #guarded: array:1 [
                            0 => "*"
                          ]
                        }
                        5 => App\Models\Term {#1143
                          #connection: "mysql"
                          #table: "terms"
                          #primaryKey: "id"
                          #keyType: "int"
                          +incrementing: false
                          #with: []
                          #withCount: []
                          +preventsLazyLoading: false
                          #perPage: 15
                          +exists: true
                          +wasRecentlyCreated: false
                          #escapeWhenCastingToString: false
                          #attributes: array:12 [
                            "id" => 5622
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Рослова"
                            "field_bio" => null
                            "field_degree_rp" => null
                            "field_foreign_lang_name" => null
                            "field_foreign_lang_patronymic" => null
                            "field_foreign_lang_surname" => null
                            "field_name" => "Лариса"
                            "field_patronymic" => "Олеговна"
                            "field_surname_rp" => null
                          ]
                          #original: array:12 [
                            "id" => 5622
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Рослова"
                            "field_bio" => null
                            "field_degree_rp" => null
                            "field_foreign_lang_name" => null
                            "field_foreign_lang_patronymic" => null
                            "field_foreign_lang_surname" => null
                            "field_name" => "Лариса"
                            "field_patronymic" => "Олеговна"
                            "field_surname_rp" => null
                          ]
                          #changes: []
                          #casts: []
                          #classCastCache: []
                          #attributeCastCache: []
                          #dateFormat: null
                          #appends: []
                          #dispatchesEvents: []
                          #observables: []
                          #relations: []
                          #touches: []
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                          #hidden: []
                          #visible: []
                          #fillable: []
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                            0 => "*"
                          ]
                        }
                      ]
                      #escapeWhenCastingToString: false
                    }
                    "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1144
                      #items: []
                      #escapeWhenCastingToString: false
                    }
                    "field_book_type" => Illuminate\Database\Eloquent\Collection {#1145
                      #items: array:1 [
                        0 => App\Models\Term {#1146
                          #connection: "mysql"
                          #table: "terms"
                          #primaryKey: "id"
                          #keyType: "int"
                          +incrementing: false
                          #with: []
                          #withCount: []
                          +preventsLazyLoading: false
                          #perPage: 15
                          +exists: true
                          +wasRecentlyCreated: false
                          #escapeWhenCastingToString: false
                          #attributes: array:10 [
                            "id" => 6671
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Учебник"
                            "field_book_type_foreign" => null
                            "field_cases" => array:6 [
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                              "field_creative_case" => "учебником"
                              "field_dative_case" => "учебнику"
                              "field_genitive_case" => "учебника"
                              "field_nominative_case" => "учебник"
                              "field_prepositional_case" => "учебнике"
                            ]
                            "field_plural_form" => null
                            "field_short_name" => null
                            "field_short_name_foreign" => null
                            "field_translit" => "uchebnik"
                          ]
                          #original: array:10 [
                            "id" => 6671
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Учебник"
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                            "field_cases" => array:6 [
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                              "field_creative_case" => "учебником"
                              "field_dative_case" => "учебнику"
                              "field_genitive_case" => "учебника"
                              "field_nominative_case" => "учебник"
                              "field_prepositional_case" => "учебнике"
                            ]
                            "field_plural_form" => null
                            "field_short_name" => null
                            "field_short_name_foreign" => null
                            "field_translit" => "uchebnik"
                          ]
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                          #attributeCastCache: []
                          #dateFormat: null
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                          #dispatchesEvents: []
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                          #relations: []
                          #touches: []
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                          ]
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                      ]
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                    }
                    "field_country" => Illuminate\Database\Eloquent\Collection {#1147
                      #items: array:1 [
                        0 => App\Models\Term {#1148
                          #connection: "mysql"
                          #table: "terms"
                          #primaryKey: "id"
                          #keyType: "int"
                          +incrementing: false
                          #with: []
                          #withCount: []
                          +preventsLazyLoading: false
                          #perPage: 15
                          +exists: true
                          +wasRecentlyCreated: false
                          #escapeWhenCastingToString: false
                          #attributes: array:6 [
                            "id" => 9
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Россия"
                            "field_cases" => array:6 [
                              "field_accusative_case" => null
                              "field_creative_case" => null
                              "field_dative_case" => null
                              "field_genitive_case" => null
                              "field_nominative_case" => null
                              "field_prepositional_case" => null
                            ]
                            "field_translit" => "rossiya"
                          ]
                          #original: array:6 [
                            "id" => 9
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Россия"
                            "field_cases" => array:6 [
                              "field_accusative_case" => null
                              "field_creative_case" => null
                              "field_dative_case" => null
                              "field_genitive_case" => null
                              "field_nominative_case" => null
                              "field_prepositional_case" => null
                            ]
                            "field_translit" => "rossiya"
                          ]
                          #changes: []
                          #casts: []
                          #classCastCache: []
                          #attributeCastCache: []
                          #dateFormat: null
                          #appends: []
                          #dispatchesEvents: []
                          #observables: []
                          #relations: []
                          #touches: []
                          +timestamps: true
                          +usesUniqueIds: false
                          #hidden: []
                          #visible: []
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                        }
                      ]
                      #escapeWhenCastingToString: false
                    }
                    "field_city" => Illuminate\Database\Eloquent\Collection {#1149
                      #items: array:1 [
                        0 => App\Models\Term {#1150
                          #connection: "mysql"
                          #table: "terms"
                          #primaryKey: "id"
                          #keyType: "int"
                          +incrementing: false
                          #with: []
                          #withCount: []
                          +preventsLazyLoading: false
                          #perPage: 15
                          +exists: true
                          +wasRecentlyCreated: false
                          #escapeWhenCastingToString: false
                          #attributes: array:6 [
                            "id" => 15
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Москва"
                            "field_cases" => array:6 [
                              "field_accusative_case" => null
                              "field_creative_case" => null
                              "field_dative_case" => null
                              "field_genitive_case" => null
                              "field_nominative_case" => null
                              "field_prepositional_case" => null
                            ]
                            "field_translit" => "moskva"
                          ]
                          #original: array:6 [
                            "id" => 15
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "Москва"
                            "field_cases" => array:6 [
                              "field_accusative_case" => null
                              "field_creative_case" => null
                              "field_dative_case" => null
                              "field_genitive_case" => null
                              "field_nominative_case" => null
                              "field_prepositional_case" => null
                            ]
                            "field_translit" => "moskva"
                          ]
                          #changes: []
                          #casts: []
                          #classCastCache: []
                          #attributeCastCache: []
                          #dateFormat: null
                          #appends: []
                          #dispatchesEvents: []
                          #observables: []
                          #relations: []
                          #touches: []
                          +timestamps: true
                          +usesUniqueIds: false
                          #hidden: []
                          #visible: []
                          #fillable: []
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                            0 => "*"
                          ]
                        }
                      ]
                      #escapeWhenCastingToString: false
                    }
                    "field_series" => Illuminate\Database\Eloquent\Collection {#1151
                      #items: array:1 [
                        0 => App\Models\Term {#1152
                          #connection: "mysql"
                          #table: "terms"
                          #primaryKey: "id"
                          #keyType: "int"
                          +incrementing: false
                          #with: []
                          #withCount: []
                          +preventsLazyLoading: false
                          #perPage: 15
                          +exists: true
                          +wasRecentlyCreated: false
                          #escapeWhenCastingToString: false
                          #attributes: array:8 [
                            "id" => 4795
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "сферы"
                            "field_cases" => null
                            "field_foreign_lang_series" => null
                            "field_transcription_eng" => null
                            "field_translit" => "sfery"
                          ]
                          #original: array:8 [
                            "id" => 4795
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "сферы"
                            "field_cases" => null
                            "field_foreign_lang_series" => null
                            "field_transcription_eng" => null
                            "field_translit" => "sfery"
                          ]
                          #changes: []
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                          #classCastCache: []
                          #attributeCastCache: []
                          #dateFormat: null
                          #appends: []
                          #dispatchesEvents: []
                          #observables: []
                          #relations: []
                          #touches: []
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                          #hidden: []
                          #visible: []
                          #fillable: []
                          #guarded: array:1 [
                            0 => "*"
                          ]
                        }
                      ]
                      #escapeWhenCastingToString: false
                    }
                    "field_umk" => Illuminate\Database\Eloquent\Collection {#1153
                      #items: []
                      #escapeWhenCastingToString: false
                    }
                    "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1154
                      #items: array:1 [
                        0 => App\Models\Term {#1155
                          #connection: "mysql"
                          #table: "terms"
                          #primaryKey: "id"
                          #keyType: "int"
                          +incrementing: false
                          #with: []
                          #withCount: []
                          +preventsLazyLoading: false
                          #perPage: 15
                          +exists: true
                          +wasRecentlyCreated: false
                          #escapeWhenCastingToString: false
                          #attributes: array:6 [
                            "id" => 38
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "базовый"
                            "field_cases" => array:6 [
                              "field_accusative_case" => "базовый"
                              "field_creative_case" => "базовым"
                              "field_dative_case" => "базовому"
                              "field_genitive_case" => "базового"
                              "field_nominative_case" => "базовый"
                              "field_prepositional_case" => "базовом"
                            ]
                            "field_translit" => "bazovyy"
                          ]
                          #original: array:6 [
                            "id" => 38
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "базовый"
                            "field_cases" => array:6 [
                              "field_accusative_case" => "базовый"
                              "field_creative_case" => "базовым"
                              "field_dative_case" => "базовому"
                              "field_genitive_case" => "базового"
                              "field_nominative_case" => "базовый"
                              "field_prepositional_case" => "базовом"
                            ]
                            "field_translit" => "bazovyy"
                          ]
                          #changes: []
                          #casts: []
                          #classCastCache: []
                          #attributeCastCache: []
                          #dateFormat: null
                          #appends: []
                          #dispatchesEvents: []
                          #observables: []
                          #relations: []
                          #touches: []
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                          +usesUniqueIds: false
                          #hidden: []
                          #visible: []
                          #fillable: []
                          #guarded: array:1 [
                            0 => "*"
                          ]
                        }
                      ]
                      #escapeWhenCastingToString: false
                    }
                    "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1156
                      #items: array:1 [
                        0 => App\Models\Term {#1157
                          #connection: "mysql"
                          #table: "terms"
                          #primaryKey: "id"
                          #keyType: "int"
                          +incrementing: false
                          #with: []
                          #withCount: []
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                          #perPage: 15
                          +exists: true
                          +wasRecentlyCreated: false
                          #escapeWhenCastingToString: false
                          #attributes: array:10 [
                            "id" => 25
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "ФГОС (новый)"
                            "field_cases" => array:6 [
                              "field_accusative_case" => null
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                              "field_dative_case" => null
                              "field_genitive_case" => null
                              "field_nominative_case" => null
                              "field_prepositional_case" => null
                            ]
                            "field_color" => "красный"
                            "field_end_year" => "2027"
                            "field_start_year" => "2023"
                            "field_translit" => "fgos"
                            "field_type" => "новый"
                          ]
                          #original: array:10 [
                            "id" => 25
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "ФГОС (новый)"
                            "field_cases" => array:6 [
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                              "field_nominative_case" => null
                              "field_prepositional_case" => null
                            ]
                            "field_color" => "красный"
                            "field_end_year" => "2027"
                            "field_start_year" => "2023"
                            "field_translit" => "fgos"
                            "field_type" => "новый"
                          ]
                          #changes: []
                          #casts: []
                          #classCastCache: []
                          #attributeCastCache: []
                          #dateFormat: null
                          #appends: []
                          #dispatchesEvents: []
                          #observables: []
                          #relations: []
                          #touches: []
                          +timestamps: true
                          +usesUniqueIds: false
                          #hidden: []
                          #visible: []
                          #fillable: []
                          #guarded: array:1 [
                            0 => "*"
                          ]
                        }
                      ]
                      #escapeWhenCastingToString: false
                    }
                    "field_publication_number" => null
                    "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1158
                      #items: []
                      #escapeWhenCastingToString: false
                    }
                    "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1159
                      #items: []
                      #escapeWhenCastingToString: false
                    }
                    "field_under_the_edition_degree" => null
                    "field_cover_description" => "с компасом"
                    "field_publication_year" => "2023"
                    "field_publication_year_until" => null
                    "field_part" => ""
                    "field_part_writing" => ""
                    "field_second_foreign_language" => null
                    "field_for_whom" => "Учебное пособие"
                    "field_allowed" => null
                    "field_reserve_field" => null
                    "field_link_to_source" => null
                    "field_tasks_count" => "941"
                    "field_priority" => "10"
                    "field_default_folder" => "/matematika_05/bunimovich-u23/"
                    "field_isbn" => "978-5-09-105798-0"
                    "field_cover" => array:1 [
                      0 => "/media/matematika_05/bunimovich-u23/covers/cover1.webp?ts=1737321399"
                    ]
                    "field_cover_alts" => array:1 [
                      0 => ""
                    ]
                    "field_covers" => array:1 [
                      0 => array:4 [
                        "path" => "/media/matematika_05/bunimovich-u23/covers/cover1.webp?ts=1737321399"
                        "alt" => ""
                        "width" => "1200"
                        "height" => "1604"
                      ]
                    ]
                    "field_popular_book" => null
                    "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1160
                      #items: []
                      #escapeWhenCastingToString: false
                    }
                    "field_new_book" => Illuminate\Database\Eloquent\Collection {#1161
                      #items: []
                      #escapeWhenCastingToString: false
                    }
                    "field_old_book" => Illuminate\Database\Eloquent\Collection {#1162
                      #items: []
                      #escapeWhenCastingToString: false
                    }
                    "field_url" => "/5-klass/matematika/bunimovich-uchebnik"
                    "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1163
                      #items: array:1 [
                        0 => App\Models\Term {#1164
                          #connection: "mysql"
                          #table: "terms"
                          #primaryKey: "id"
                          #keyType: "int"
                          +incrementing: false
                          #with: []
                          #withCount: []
                          +preventsLazyLoading: false
                          #perPage: 15
                          +exists: true
                          +wasRecentlyCreated: false
                          #escapeWhenCastingToString: false
                          #attributes: array:6 [
                            "id" => 50
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "оранжевый"
                            "field_cases" => array:6 [
                              "field_accusative_case" => "оранжевый"
                              "field_creative_case" => "оранжевым"
                              "field_dative_case" => "оранжевому"
                              "field_genitive_case" => "оранжевого"
                              "field_nominative_case" => "оранжевый"
                              "field_prepositional_case" => "оранжевом"
                            ]
                            "field_translit" => "oranzhevyy"
                          ]
                          #original: array:6 [
                            "id" => 50
                            "created_at" => "2026-04-10 13:58:26"
                            "updated_at" => null
                            "name" => "оранжевый"
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                              "field_genitive_case" => "оранжевого"
                              "field_nominative_case" => "оранжевый"
                              "field_prepositional_case" => "оранжевом"
                            ]
                            "field_translit" => "oranzhevyy"
                          ]
                          #changes: []
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                          #classCastCache: []
                          #attributeCastCache: []
                          #dateFormat: null
                          #appends: []
                          #dispatchesEvents: []
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                          #relations: []
                          #touches: []
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                          +usesUniqueIds: false
                          #hidden: []
                          #visible: []
                          #fillable: []
                          #guarded: array:1 [
                            0 => "*"
                          ]
                        }
                      ]
                      #escapeWhenCastingToString: false
                    }
                    "field_metatags_title" => null
                    "field_metatags_description" => null
                    "field_h1" => null
                    "field_description_top" => null
                    "field_description_bottom" => null
                    "breadcrumbs" => array:3 [
                      "class" => 376
                      "subject" => 544
                      "class_subject" => 398
                    ]
                  ]
                  #original: array:50 [
                    "id" => 59
                    "created_at" => "2026-04-10 13:58:26"
                    "updated_at" => null
                    "field_tree_default_status" => "tasks"
                    "field_pages_status" => null
                    "field_subject" => Illuminate\Database\Eloquent\Collection {#1132}
                    "field_class" => Illuminate\Database\Eloquent\Collection {#1133}
                    "field_publisher" => Illuminate\Database\Eloquent\Collection {#1135}
                    "field_author" => Illuminate\Database\Eloquent\Collection {#1137}
                    "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1144}
                    "field_book_type" => Illuminate\Database\Eloquent\Collection {#1145}
                    "field_country" => Illuminate\Database\Eloquent\Collection {#1147}
                    "field_city" => Illuminate\Database\Eloquent\Collection {#1149}
                    "field_series" => Illuminate\Database\Eloquent\Collection {#1151}
                    "field_umk" => Illuminate\Database\Eloquent\Collection {#1153}
                    "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1154}
                    "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1156}
                    "field_publication_number" => null
                    "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1158}
                    "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1159}
                    "field_under_the_edition_degree" => null
                    "field_cover_description" => "с компасом"
                    "field_publication_year" => "2023"
                    "field_publication_year_until" => null
                    "field_part" => ""
                    "field_part_writing" => ""
                    "field_second_foreign_language" => null
                    "field_for_whom" => "Учебное пособие"
                    "field_allowed" => null
                    "field_reserve_field" => null
                    "field_link_to_source" => null
                    "field_tasks_count" => "941"
                    "field_priority" => "10"
                    "field_default_folder" => "/matematika_05/bunimovich-u23/"
                    "field_isbn" => "978-5-09-105798-0"
                    "field_cover" => array:1 [
                      0 => "/media/matematika_05/bunimovich-u23/covers/cover1.webp?ts=1737321399"
                    ]
                    "field_cover_alts" => array:1 [
                      0 => ""
                    ]
                    "field_covers" => array:1 [
                      0 => array:4 [
                        "path" => "/media/matematika_05/bunimovich-u23/covers/cover1.webp?ts=1737321399"
                        "alt" => ""
                        "width" => "1200"
                        "height" => "1604"
                      ]
                    ]
                    "field_popular_book" => null
                    "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1160}
                    "field_new_book" => Illuminate\Database\Eloquent\Collection {#1161}
                    "field_old_book" => Illuminate\Database\Eloquent\Collection {#1162}
                    "field_url" => "/5-klass/matematika/bunimovich-uchebnik"
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            "text" => "<p><strong>ВОПРОСЫ И ЗАДАНИЯ</strong></p><p><strong>Какое число называется простым; составным? Какое число не относится ни к одному из этих двух видов?</strong></p><p><strong>Перечислите в порядке возрастания первые десять простых чисел.</strong></p><p><strong>Простым или составным является число:</strong></p><p>а) 67; б) 91; в) 479; г) 869?</p>"
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                "path" => "/media/matematika_05/bunimovich-u23/0-vop/21-1.webp?ts=1737374597"
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              "type" => "task"
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            "text" => "<p><strong>ВОПРОСЫ И ЗАДАНИЯ</strong></p><p><strong>Какое число называется простым; составным? Какое число не относится ни к одному из этих двух видов?</strong></p><p><strong>Перечислите в порядке возрастания первые десять простых чисел.</strong></p><p><strong>Простым или составным является число:</strong></p><p>а) 67; б) 91; в) 479; г) 869?</p>"
            "img" => array:1 [
              0 => array:5 [
                "name" => "21-1.jpg"
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                "width" => "769"
                "height" => 723
                "path" => "/media/matematika_05/bunimovich-u23/0-vop/21-1.webp?ts=1737374597"
              ]
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          ]
        }
        1 => App\Models\Element {#1102
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            "text" => "<p><strong>Какое число называется простым; составным? Какое число не относится ни к одному из этих двух видов?</strong><br> Натуральное число (т.е. целое положительное число, используемое при счете предметов) называется <b>простым</b>, если оно больше 1 и имеет ровно два различных натуральных делителя: единицу и само себя.<br> Натуральное число называется <b>составным</b>, если оно больше 1 и не является простым, то есть у него есть другие делители, кроме 1 и самого себя. Иными словами, у составного числа более двух делителей.<br> Число <b>1</b> (единица) является особым случаем. Оно не относится ни к простым, ни к составным числам, так как у него всего один натуральный делитель — само число 1.<br> Ответ: Простое число — это натуральное число больше 1, имеющее ровно два делителя (1 и само себя). Составное число — это натуральное число больше 1, имеющее более двух делителей. Число 1 не является ни простым, ни составным.</p><p><strong>Перечислите в порядке возрастания первые десять простых чисел.</strong><br> Для нахождения первых десяти простых чисел будем последовательно рассматривать натуральные числа, начиная с 2, и определять, являются ли они простыми.<br> - 2 — простое (делители 1, 2). Это единственное четное простое число.<br> - 3 — простое (делители 1, 3).<br> - 4 = 2 &times; 2 — составное.<br> - 5 — простое (делители 1, 5).<br> - 6 = 2 &times; 3 — составное.<br> - 7 — простое (делители 1, 7).<br> - 8, 9, 10 — составные.<br> - 11 — простое.<br> - 13 — простое.<br> - 17 — простое.<br> - 19 — простое.<br> - 23 — простое.<br> - 29 — простое.<br> Таким образом, мы нашли первые десять простых чисел.<br> Ответ: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.</p><p><strong>Простым или составным является число: а) 67; б) 91; в) 479; г) 869?</strong><br> Чтобы определить, является ли число простым, нужно проверить его делимость на простые числа, не превосходящие квадратный корень из этого числа. Если ни одного такого делителя не найдется, число является простым.<br> <b>а) 67:</b><br> Найдём $\sqrt{67}$. Так как $8^2 = 64$ и $9^2 = 81$, то $\sqrt{67} \approx 8.18$. Необходимо проверить делимость на простые числа, которые меньше 8.18: это 2, 3, 5, 7.<br> - 67 не делится на 2 (так как число нечетное).<br> - 67 не делится на 3 (так как сумма цифр $6+7=13$ не делится на 3).<br> - 67 не делится на 5 (так как не оканчивается на 0 или 5).<br> - При делении на 7 получаем: $67 = 7 \times 9 + 4$. Не делится.<br> Так как 67 не имеет простых делителей до $\sqrt{67}$, оно является простым.<br> <b>б) 91:</b><br> Найдём $\sqrt{91}$. Так как $9^2 = 81$ и $10^2 = 100$, то $\sqrt{91} \approx 9.54$. Проверяем делимость на простые числа: 2, 3, 5, 7.<br> - 91 не делится на 2, 3, 5.<br> - Проверяем делимость на 7: $91 \div 7 = 13$.<br> Число 91 имеет делители 7 и 13, следовательно, оно является составным.<br> <b>в) 479:</b><br> Найдём $\sqrt{479}$. Так как $21^2 = 441$ и $22^2 = 484$, то $\sqrt{479} \approx 21.88$. Проверяем делимость на простые числа до 21: 2, 3, 5, 7, 11, 13, 17, 19.<br> - 479 не делится на 2, 3, 5.<br> - $479 \div 7 = 68$ (остаток 3).<br> - Признак делимости на 11: знакопеременная сумма цифр $(9 - 7 + 4) = 6$ не делится на 11.<br> - $479 \div 13 = 36$ (остаток 11).<br> - $479 \div 17 = 28$ (остаток 3).<br> - $479 \div 19 = 25$ (остаток 4).<br> Делителей не найдено, следовательно, 479 — простое число.<br> <b>г) 869:</b><br> Найдём $\sqrt{869}$. Так как $29^2 = 841$ и $30^2 = 900$, то $\sqrt{869} \approx 29.48$. Проверяем делимость на простые числа до 29.<br> - Воспользуемся признаком делимости на 11: знакопеременная сумма цифр должна делиться на 11. Для 869: $(9 - 6 + 8) = 11$. Так как 11 делится на 11, то и 869 делится на 11.<br> - Найдём результат деления: $869 \div 11 = 79$.<br> Число 869 имеет делители 11 и 79, следовательно, оно является составным.<br> Ответ: а) 67 — простое; б) 91 — составное; в) 479 — простое; г) 869 — составное.</p>"
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            "text" => "<p><strong>Какое число называется простым; составным? Какое число не относится ни к одному из этих двух видов?</strong><br> Натуральное число (т.е. целое положительное число, используемое при счете предметов) называется <b>простым</b>, если оно больше 1 и имеет ровно два различных натуральных делителя: единицу и само себя.<br> Натуральное число называется <b>составным</b>, если оно больше 1 и не является простым, то есть у него есть другие делители, кроме 1 и самого себя. Иными словами, у составного числа более двух делителей.<br> Число <b>1</b> (единица) является особым случаем. Оно не относится ни к простым, ни к составным числам, так как у него всего один натуральный делитель — само число 1.<br> Ответ: Простое число — это натуральное число больше 1, имеющее ровно два делителя (1 и само себя). Составное число — это натуральное число больше 1, имеющее более двух делителей. Число 1 не является ни простым, ни составным.</p><p><strong>Перечислите в порядке возрастания первые десять простых чисел.</strong><br> Для нахождения первых десяти простых чисел будем последовательно рассматривать натуральные числа, начиная с 2, и определять, являются ли они простыми.<br> - 2 — простое (делители 1, 2). Это единственное четное простое число.<br> - 3 — простое (делители 1, 3).<br> - 4 = 2 &times; 2 — составное.<br> - 5 — простое (делители 1, 5).<br> - 6 = 2 &times; 3 — составное.<br> - 7 — простое (делители 1, 7).<br> - 8, 9, 10 — составные.<br> - 11 — простое.<br> - 13 — простое.<br> - 17 — простое.<br> - 19 — простое.<br> - 23 — простое.<br> - 29 — простое.<br> Таким образом, мы нашли первые десять простых чисел.<br> Ответ: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.</p><p><strong>Простым или составным является число: а) 67; б) 91; в) 479; г) 869?</strong><br> Чтобы определить, является ли число простым, нужно проверить его делимость на простые числа, не превосходящие квадратный корень из этого числа. Если ни одного такого делителя не найдется, число является простым.<br> <b>а) 67:</b><br> Найдём $\sqrt{67}$. Так как $8^2 = 64$ и $9^2 = 81$, то $\sqrt{67} \approx 8.18$. Необходимо проверить делимость на простые числа, которые меньше 8.18: это 2, 3, 5, 7.<br> - 67 не делится на 2 (так как число нечетное).<br> - 67 не делится на 3 (так как сумма цифр $6+7=13$ не делится на 3).<br> - 67 не делится на 5 (так как не оканчивается на 0 или 5).<br> - При делении на 7 получаем: $67 = 7 \times 9 + 4$. Не делится.<br> Так как 67 не имеет простых делителей до $\sqrt{67}$, оно является простым.<br> <b>б) 91:</b><br> Найдём $\sqrt{91}$. Так как $9^2 = 81$ и $10^2 = 100$, то $\sqrt{91} \approx 9.54$. Проверяем делимость на простые числа: 2, 3, 5, 7.<br> - 91 не делится на 2, 3, 5.<br> - Проверяем делимость на 7: $91 \div 7 = 13$.<br> Число 91 имеет делители 7 и 13, следовательно, оно является составным.<br> <b>в) 479:</b><br> Найдём $\sqrt{479}$. Так как $21^2 = 441$ и $22^2 = 484$, то $\sqrt{479} \approx 21.88$. Проверяем делимость на простые числа до 21: 2, 3, 5, 7, 11, 13, 17, 19.<br> - 479 не делится на 2, 3, 5.<br> - $479 \div 7 = 68$ (остаток 3).<br> - Признак делимости на 11: знакопеременная сумма цифр $(9 - 7 + 4) = 6$ не делится на 11.<br> - $479 \div 13 = 36$ (остаток 11).<br> - $479 \div 17 = 28$ (остаток 3).<br> - $479 \div 19 = 25$ (остаток 4).<br> Делителей не найдено, следовательно, 479 — простое число.<br> <b>г) 869:</b><br> Найдём $\sqrt{869}$. Так как $29^2 = 841$ и $30^2 = 900$, то $\sqrt{869} \approx 29.48$. Проверяем делимость на простые числа до 29.<br> - Воспользуемся признаком делимости на 11: знакопеременная сумма цифр должна делиться на 11. Для 869: $(9 - 6 + 8) = 11$. Так как 11 делится на 11, то и 869 делится на 11.<br> - Найдём результат деления: $869 \div 11 = 79$.<br> Число 869 имеет делители 11 и 79, следовательно, оно является составным.<br> Ответ: а) 67 — простое; б) 91 — составное; в) 479 — простое; г) 869 — составное.</p>"
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Вопросы и задания (с. 101)

Просмотр Редак-ть Редак-ть (переопр)

Условие. Вопросы и задания (с. 101)

Редак-ть Редакция

скриншот условия

ВОПРОСЫ И ЗАДАНИЯ

Какое число называется простым; составным? Какое число не относится ни к одному из этих двух видов?

Перечислите в порядке возрастания первые десять простых чисел.

Простым или составным является число:

а) 67; б) 91; в) 479; г) 869?

Решение 2. Вопросы и задания (с. 101)

Редак-ть Редакция

Какое число называется простым; составным? Какое число не относится ни к одному из этих двух видов?
Натуральное число (т.е. целое положительное число, используемое при счете предметов) называется простым, если оно больше 1 и имеет ровно два различных натуральных делителя: единицу и само себя.
Натуральное число называется составным, если оно больше 1 и не является простым, то есть у него есть другие делители, кроме 1 и самого себя. Иными словами, у составного числа более двух делителей.
Число 1 (единица) является особым случаем. Оно не относится ни к простым, ни к составным числам, так как у него всего один натуральный делитель — само число 1.
Ответ: Простое число — это натуральное число больше 1, имеющее ровно два делителя (1 и само себя). Составное число — это натуральное число больше 1, имеющее более двух делителей. Число 1 не является ни простым, ни составным.

Перечислите в порядке возрастания первые десять простых чисел.
Для нахождения первых десяти простых чисел будем последовательно рассматривать натуральные числа, начиная с 2, и определять, являются ли они простыми.
- 2 — простое (делители 1, 2). Это единственное четное простое число.
- 3 — простое (делители 1, 3).
- 4 = 2 × 2 — составное.
- 5 — простое (делители 1, 5).
- 6 = 2 × 3 — составное.
- 7 — простое (делители 1, 7).
- 8, 9, 10 — составные.
- 11 — простое.
- 13 — простое.
- 17 — простое.
- 19 — простое.
- 23 — простое.
- 29 — простое.
Таким образом, мы нашли первые десять простых чисел.
Ответ: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

Простым или составным является число: а) 67; б) 91; в) 479; г) 869?
Чтобы определить, является ли число простым, нужно проверить его делимость на простые числа, не превосходящие квадратный корень из этого числа. Если ни одного такого делителя не найдется, число является простым.
а) 67:
Найдём $\sqrt{67}$. Так как $8^2 = 64$ и $9^2 = 81$, то $\sqrt{67} \approx 8.18$. Необходимо проверить делимость на простые числа, которые меньше 8.18: это 2, 3, 5, 7.
- 67 не делится на 2 (так как число нечетное).
- 67 не делится на 3 (так как сумма цифр $6+7=13$ не делится на 3).
- 67 не делится на 5 (так как не оканчивается на 0 или 5).
- При делении на 7 получаем: $67 = 7 \times 9 + 4$. Не делится.
Так как 67 не имеет простых делителей до $\sqrt{67}$, оно является простым.
б) 91:
Найдём $\sqrt{91}$. Так как $9^2 = 81$ и $10^2 = 100$, то $\sqrt{91} \approx 9.54$. Проверяем делимость на простые числа: 2, 3, 5, 7.
- 91 не делится на 2, 3, 5.
- Проверяем делимость на 7: $91 \div 7 = 13$.
Число 91 имеет делители 7 и 13, следовательно, оно является составным.
в) 479:
Найдём $\sqrt{479}$. Так как $21^2 = 441$ и $22^2 = 484$, то $\sqrt{479} \approx 21.88$. Проверяем делимость на простые числа до 21: 2, 3, 5, 7, 11, 13, 17, 19.
- 479 не делится на 2, 3, 5.
- $479 \div 7 = 68$ (остаток 3).
- Признак делимости на 11: знакопеременная сумма цифр $(9 - 7 + 4) = 6$ не делится на 11.
- $479 \div 13 = 36$ (остаток 11).
- $479 \div 17 = 28$ (остаток 3).
- $479 \div 19 = 25$ (остаток 4).
Делителей не найдено, следовательно, 479 — простое число.
г) 869:
Найдём $\sqrt{869}$. Так как $29^2 = 841$ и $30^2 = 900$, то $\sqrt{869} \approx 29.48$. Проверяем делимость на простые числа до 29.
- Воспользуемся признаком делимости на 11: знакопеременная сумма цифр должна делиться на 11. Для 869: $(9 - 6 + 8) = 11$. Так как 11 делится на 11, то и 869 делится на 11.
- Найдём результат деления: $869 \div 11 = 79$.
Число 869 имеет делители 11 и 79, следовательно, оно является составным.
Ответ: а) 67 — простое; б) 91 — составное; в) 479 — простое; г) 869 — составное.