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Здравствуйте, lyskov.kirill! Тут все просто. попробуйте подставить в Вашу формулу n=1 получиться 10+6-3-1=12 которое на 63 не делиться, а поскольку n=1 является натуральным числом, значит не для всех натуральных выполняется делимость на 63. Если взять n=2 то получиться в результате 126, оно будет делиться на 63, а вот уже при n=3 опять получиться число, которое не будет делиться на 63
В ответе написано, что делится при всех четных n и не делится при всех нечетных. Мне нужно обоснованное решение. Я так понимаю это доказывается через индукцию, а у меня с этим плохо.
Тогда условие задачи должно звучать несколько иначе, потому как фраза n принадлежит N означает все натуральные числа вот чего получилось.. легче не придумала
Чтобы число делилось на 63, оно должно делиться на 9 и на 7. Делимость на 9. Очевидно, 10^n - 1 делится на 9 при любом n = 1,2, ... . 6^n и 3^n при n >= 2 имеют по крайней мере две тройки в разложении на простые множители, и, следовательно, делятся на 9. Значит, 10^n + 6^n - 3^n - 1 делится на 9 при n>= 2. Делимость на 7. 10^n = (7 + 3)^n. Как несложно понять, отсюда следует, что 10^n и 3^n дают одинаковые остатки при делении на 7. Поэтому 10^n - 3^n делится на 7 при всех натуральных n. 6^n = (7 - 1)^n. При n = 1, 2, 3, ... деление на 7 даст остатки -1, 1, -1, 1, ... (или, что тоже, 6, 1, 6, 1, ... ). Отсюда следует, что 6^n - 1 делится на 7 при четных n и не делится при нечетных. Подводя итог, получим, что 10^n + 6^n - 3^n - 1 делится на 63 при четных n.
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