[Math] Finding the remainder when $2^{100}+3^{100}+4^{100}+5^{100}$ is divided by $7$

Question

Find the remainder when $2^{100}+3^{100}+4^{100}+5^{100}$ is divided by $7$.

Please brief about the concept behind this to solve such problems. Thanks.

Answer 1

The idea is to reach at $\equiv\pm1\pmod n$ for a given modulo integer $n$

In general we should utilize Fermat's little theorem for prime modulo

or Euler's Totient Theorem or Carmichael Function for non-prime modulo

unless we can reach $\pm1$ easily like below.

$2^3=8\equiv1\pmod 7\implies 2^{100}=2\cdot (2^3)^{33}\equiv2\cdot 1^{33}\pmod 7\equiv2$

$4^3=64\equiv1\pmod 7\implies 4^{100}=4\cdot (4^3)^{33}\equiv4\cdot1^{33}\pmod 7\equiv4$

$3^3=27\equiv-1\pmod 7\implies 3^{100}=3\cdot (3^3)^{33}\equiv3\cdot(-1)^{33}\pmod 7\equiv-3$

$5^3=125\equiv-1\pmod 7\implies 5^{100}=5\cdot (5^3)^{33}\equiv5\cdot(-1)^{33}\pmod 7\equiv-5$