Предмет: Алгебра, автор: ulukzumabaev9

Найдите две последние цифры числа 7^2023

Ответы

Автор ответа: Mrdenk

9

Запишем первые несколько степений числа 7

7^1 = 7 (две последние цифры 07)

7^2 = 49 (две последние цифры 49)

7^3 = 343 (две последние цифры 43)

7^4 = 2401 (две последние цифры 01)

7^5 = 16807 (две последние цифры 07)

Заметим, что у нас обрадовался цикл.

7^(1 + 4k, где k - целое неотрицательное число) будет иметь на конце две цифры 07 (так как 7^1 заканчивается на 07 и 7^5 заканчивается на 07, а разница между показателями их степени 5 - 1 = 4)

Найдём ближайшую степень к 2023, которая заканчивается на 07

1 + 4k <= 2023

4k <= 2022

k <= 505.5

Так как k - целое неотрицательное число, k = 505

Значит 7^(1 + 4 * 505) = 7^(2021) оканчивается на 07, тогда из первых степеней числа 7, записанных выше

7^2022 оканчивается на 49

7^2023 оканчивается на 43

Ответ: 43

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