UPSC CDS 1 2021 Math Paper

Concept:The unit digit of a power depends only on the unit digit of the base.For base $7$, the unit digit repeats in a cycle of $4$: $7, 9, 3, 1$.Explanation:The unit digit of $67^{32}$ is the same as the unit digit of $7^{32}$.Observe the cycle of unit digits for powers of $7$:$7^1 \to 7$, $7^2 \to 9$, $7^3 \to 3$, $7^4 \to 1$.The cycle length is $4$.Since $32$ is a multiple of $4$ ($32 \div 4 = 8$), the unit digit of $7^{32}$ is the same as that of $7^4$, which is $1$.Alternatively, $7^{32} = (7^4)^8 = 2401^8$, and $2401$ ends with $1$, so any power of $2401$ also ends with $1$.Answer:$1$ (Option A).