Question

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

Answer 1

The correct answer is 7
(2023)2023
=(2030−7)2023
=(35K−7)2023
=2023C0​(35K)2023(−7)0+2023C1​(35K)2022(−7)+….+
…⋯+2023C2022​(−7)2023
=35N−72023.
Now ,−72023=−7×72022=−7(72)1011
=−7(50−1)1011
=−7(1011C0​501011−1011C1​(50)1010+……⋅1011C1011​)
=−7(5λ−1)
=−35λ+7
∴ when (2023)2023 is divided by 35 remainder is 7