Find remainder when $1^{5} + 2^{5} \cdots +100^{5}$ divided by 4 I'm studding D.M Burton & want to solve: Find remainder when $1^{5} + 2^{5} \cdots +100^{5}$ divided by $4$. . Please help me by giving your solution to it. I'm new comer to number theory so please don't use theorems above Theory of Congruence.

Apply "$\bmod4$" on each term, then you get:

$\sum\limits_{n=1}^{100}n^5\equiv25\sum\limits_{n=1}^{4}(n\bmod4)^5\equiv25(1^5+2^5+3^5+4^5)\equiv32500\equiv0\pmod4$