Question

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

Answer 1

Answer: 7

$(2023{)}^{2023}$
$=(2030-7{)}^{2023}$
$=(35K-7{)}^{2023}$
$={}^{2023}{C}_0(35K{)}^{2023}(-7{)}^0+{}^{2023}{C}_1(35K{)}^{2022}(-7)+\dots .+$
$\dots \cdots +{}^{2023}{C}_{2022}(-7{)}^{2023}$
$=35N-7^{2023}.$
$\text{ Now },-7^{2023}=-7\times 7^{2022}=-7{\left(7^2\right)}^{1011}$
$=-7(50-1{)}^{1011}$
$=-7\left({}^{1011}{C}_{0}50^{1011}-{}^{1011}{C}_1(50{)}^{1010}+\dots \dots \cdot {}^{1011}{C}_{1011}\right)$
$=-7(5\lambda -1)$
$=-35\lambda +7$
$\therefore$ when $(2023{)}^{2023}$ is divided by 35 remainder is 7