Doubt concerning Fermat's little theorem with a nonprime number

Question

I'm trying to find the residue of $\frac{64^{82}}{12}$. This means that I need to find $m$ such that $64^{82} ≡_{12} m$. Using Fermat's little theorem, I have $6^11 ≡_{12} 1$. However, 12 is a nonprime number, so my question is: how can I find that residue using Fermat's little theorem?

Answer 1

$64 \cong 4 \mod 12$. $4^2 = 16 \cong 4 \mod 12$. Thus all (positive) powers of $4$ and hence $64$ are congruent to $4$ modulo $12$.