The remainder when (2023)^ 2023 is divided by 35 is ........... - Vidyadip

Question

The remainder when $(2023)^{2023}$ is divided by $35$ is $..........$.

✓

Answer

a
$(2023)^{2023}$

$=(2030-7)^{2023}$

$=(35 K -7)^{2023}$

$={ }^{2023} C _0(35 K )^{2023}(-7)^0+{ }^{2023} C _1(35 K )^{2022}(-7)+\ldots .+$

$\ldots \ldots+{ }^{2023} C _{2023}(-7)^{2023}$

$=35 N -7^{2023}$

$\text { Now, }-7^{2023}=-7 \times 7^{2022}=-7\left(7^2\right)^{1011}$

$=-7(50-1)^{1011}$

$=-7\left({ }^{1011} C _0 50^{1011}-{ }^{1011} C _1(50)^{1010}+\ldots \ldots .{ }^{1011} C _{1011}\right)$

$=-7(5 \lambda-1)$

$=-35 \lambda+7$

$\therefore$ when $(2023)^{2023}$ is divided by $35$ remainder is $7$

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