<s>[INST] 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?
 [/INST] As $3\equiv2^7\pmod{125}$
$$2^n+3^n\equiv2^n+(2^7)^n\equiv2^n(2^{6n}+1)\pmod{125}$$
So, we need $2^{6n}\equiv-1\pmod{125}$
Using Show that $r$ is a primitive root (mod $p^k$), $2$ is a primitive root $5^n$ for $n\ge1$
$$\implies6n\equiv\dfrac{\phi(125)}2\pmod{\phi(125)}$$
$$\iff3n\equiv25\pmod{50}$$
As $17\cdot3\equiv1\pmod{50}$
$$ n\equiv25\cdot17\pmod{50}\equiv25$$
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