Question

The sum of last three digits of $7^{100}-3^{100}$ is

Answer 1

Given
$7^{100}-3^{100}=(5+2{)}^{100}-(5-2{)}^{100}=2[{}^{100}{C}_1{5}^{99}\cdot 2+{}^{100}{C}_3{5}^{97}\cdot 2^3+\dots +{}^{100}{C}_{99}5\cdot 2^{99}]=2000\left[k\right],k\in N$
Therefore, the last three digits are $000$, so the sum is $0$.