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Консультация № 202231
27.02.2022, 00:28
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Уважаемые эксперты, специалисты, математики, доброго времени суток! Прошу помощи в следующем вопросе:
Задача: Натуральное число X при делении на 13 даёт остаток 7. Какой остаток при делении на 13 будет давать число Х^2 - 2 * Х?
Попытка решения: Вероятно, что Х = 13 * k + 7.
Тогда Х^2 - 2 * Х = (13 * k + 7)^2 - 2 * (13 * k + 7) = 169 * k^2 + 182 * k + 49 - 26 * k - 14 = 169 * k^2 + 156 * k + 35.
Но, что делать дальше - не знаю?
Однако ответ будет = 9, т.к. если k = 1, то 13 * 1 + 7 = 20 = X, тогда Х^2 - 2 * Х = 360 и 360 : 13 = 27 и остаток = 9!
Задача: Натуральное число X при делении на 13 даёт остаток 7. Какой остаток при делении на 13 будет давать число Х^2 - 2 * Х?
Попытка решения: Вероятно, что Х = 13 * k + 7.
Тогда Х^2 - 2 * Х = (13 * k + 7)^2 - 2 * (13 * k + 7) = 169 * k^2 + 182 * k + 49 - 26 * k - 14 = 169 * k^2 + 156 * k + 35.
Но, что делать дальше - не знаю?
Однако ответ будет = 9, т.к. если k = 1, то 13 * 1 + 7 = 20 = X, тогда Х^2 - 2 * Х = 360 и 360 : 13 = 27 и остаток = 9!
Обсуждение
27.02.2022, 04:17
общий
это ответ
Вы сделали всё правильно, остался последний шаг - разделить получившееся выражение для X[sup]2[/sup]-2X на 13. Очевидно, что 169k[sup]2[/sup]+156k+35 = 13(13k[sup]2[/sup]+12k+2)+9 при делении на 13 даст частное 13k[sup]2[/sup]+12k+2 и остаток 9 при любом целом k.
5
27.02.2022, 05:31
общий
Адресаты:
Вы спрашивали : "что делать дальше …?" - делать то, что требуется в задании: получить Остаток от деления полученного Вами числа
169·k2 + 156·k + 35 на 13 .
Остаток от деления суммы равен сумме остатков от деления всех слагаемых. Значит:
Mod(169·k2 + 156·k + 35 , 13) = Mod(169·k2 , 13) + Mod(156·k , 13) + Mod(35 , 13) = 0 + 0 + Mod(35 , 13) = 9
Множители 169 и 156 делятся на 13 БЕЗ остатка, поэтому остатки Mod(169·k2 , 13) = 0 и Mod(156·k , 13) = 0.
А остаток от деления 35 / 13 равен 9.
169·k2 + 156·k + 35 на 13 .
Остаток от деления суммы равен сумме остатков от деления всех слагаемых. Значит:
Mod(169·k2 + 156·k + 35 , 13) = Mod(169·k2 , 13) + Mod(156·k , 13) + Mod(35 , 13) = 0 + 0 + Mod(35 , 13) = 9
Множители 169 и 156 делятся на 13 БЕЗ остатка, поэтому остатки Mod(169·k2 , 13) = 0 и Mod(156·k , 13) = 0.
А остаток от деления 35 / 13 равен 9.
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