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  4. The remainder when (2021)2023 is divided by 7 is :

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Q. The remainder when $(2021)^{2023}$ is divided by $7$ is :

JEE MainJEE Main 2022Binomial Theorem

Solution:

$(2021)^{2023}=(7 \lambda-2)^{2023}$
$={ }^{2023} C _{0}(7 A )^{2023}-\ldots .{ }^{2023} C _{2023} 2^{2023}$
$=7 t -2^{2023}$
$\therefore-2^{2023}=-2 \times 2^{2022}$
$=-2 \times\left(2^{3}\right)^{674}$
$=-2(1+7 \mu)^{674}$
$=-(7 \alpha+2)$
$\Rightarrow$ remainder $=-2$ or $+5$