Задача 68251 Найти остаток ...

639064b579cc301cea21ac78

12.01.2023 12:38:43

Найти остаток

5f3ea7e3faf909182968ddd9

12.01.2023 13:21:29

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[m]10^{2022}-2^{2021}=(2\cdot 5)^{2022}-2^{2021}=2^{2021}\cdot (5^{2022}\cdot 2-1)[/m]

[m]\frac{10^{2022}-2^{2021}}{24}=\frac{2^{2021}\cdot (5^{2022}\cdot 2-1)}{2^3\cdot 3}=2^{2021-3}\cdot \frac{ (5^{2022}\cdot 2-1)}{3}=[/m]

[m]5^{2022}=25^{1011}[/m]

[m]25\equiv 1[/m](mod3)

[m]5^{2022}\cdot 2\equiv 2[/m](mod3)

[m]5^{2022}\cdot 2-1\equiv 1[/m](mod3)

О т в е т. [b]1[/b]