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Консультация № 188944
15.03.2016, 16:12
0.00 руб.
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4
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Здравствуйте! Прошу помощи в следующем вопросе:
Найти максимальное натуральное число n такое, что n^7+1 делится на n+7.
Найти максимальное натуральное число n такое, что n^7+1 делится на n+7.
Обсуждение
15.03.2016, 17:11
общий
Адресаты:
При делении многочленов получился остаток 1-77
Наибольшее n, при котором оно делится, должно быть
77-1=n+7
n=77-8
Наибольшее n, при котором оно делится, должно быть
77-1=n+7
n=77-8
16.03.2016, 09:05
общий
Адресаты:
У меня получилось 823535. Но я теорию чисел не изучал, поэтому решал по-своему.
Об авторе:
Facta loquuntur.
Facta loquuntur.
16.03.2016, 10:14
общий
это ответ
Здравствуйте, Асмик Гаряка!
Путём деления многочленов получаем
n7+1=(n+7)(n6-7n5+72n4-73n3+74n2-75n+76)+1-77
Для того, чтобы n7+1 делилось на n+7, необходимо, чтобы остаток 1-77 делился на n+7.
Наибольшее n, для которого это выполняется
1-77=-1(n+7)
77-1=n+7
n=77-8=823535
Путём деления многочленов получаем
n7+1=(n+7)(n6-7n5+72n4-73n3+74n2-75n+76)+1-77
Для того, чтобы n7+1 делилось на n+7, необходимо, чтобы остаток 1-77 делился на n+7.
Наибольшее n, для которого это выполняется
1-77=-1(n+7)
77-1=n+7
n=77-8=823535
16.03.2016, 10:50
общий
это ответ
Здравствуйте, Асмик Александровна!
При разложении двучлена n[sup]7[/sup]+1 по степеням двучлена n+1 с использованием формулы Тейлора сразу получим, что свободный член этого разложения равен (-7)[sup]7[/sup]+1=-823542. Наибольший положительный делитель свободного члена равен 823542. Тогда, учитывая, что свободный член как слагаемое суммы, которая делится на n+7 и все слагаемые которой делятся на n+7, тоже делится на n+7, и записав
получим n=823542-7=823535=7[sup]7[/sup]-8.
С уважением.
При разложении двучлена n[sup]7[/sup]+1 по степеням двучлена n+1 с использованием формулы Тейлора сразу получим, что свободный член этого разложения равен (-7)[sup]7[/sup]+1=-823542. Наибольший положительный делитель свободного члена равен 823542. Тогда, учитывая, что свободный член как слагаемое суммы, которая делится на n+7 и все слагаемые которой делятся на n+7, тоже делится на n+7, и записав
823542=1*823542=1*(n+7),
получим n=823542-7=823535=7[sup]7[/sup]-8.
С уважением.
5
Об авторе:
Facta loquuntur.
Facta loquuntur.
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