Originally Posted by

**spoon737** I have to use congruence theory to prove $\displaystyle 7|5^{2n}+3(2^{5n-2})$. I know this is just a matter of showing that $\displaystyle 5^{2n}+3(2^{5n-2}) \equiv 0 (mod 5)$, but it's arriving at that result that's giving me trouble. By the way, this problem shows up in the section of my book where congruences are introduced, and I can only use basic properties of congruences in my proof. So, if there is some nice theorem that proves this quickly, chances are I'm not allowed to use it.