Smallest $n$ such that $125|2^n+3^n$ 
Find the smallest positive integer $n$ such that $125|2^n+3^n$  

Suppose $125|2^n+3^n$. Now, $n$ must be odd because if $n$ is even then
$$0\equiv 2^n+3^n\equiv2^n+(-2)^n\equiv 2^n+2^n\equiv2^{n+1}\pmod 5,$$
which is in contrast to hypothesis. How to go on from here?

Using Proof of $a^n+b^n$ divisible by a+b when n is odd,
$$2^n+3^n$$ will be divisible by $2+3$ iff $n$ is odd $=2m+1$(say)
$$2^{2m+1}+3^{2m+1}=2^{2m+1}+(5-2)^{2m+1}$$
$$\equiv\binom{2m+1}15\cdot2^{2m}-\binom{2m+1}25^2\cdot2^{2m-1}\pmod{5^3}$$
So, we need
$$25\mid2^{2m-1}(2m+1)(2-5m)$$ 
$$\iff25\mid(2m+1)$$ as $5\nmid2^{2m-1}(2-5m)$
