modular arithmetic - Prove that $ 16^{20}+29^{21}+42^{22}$ is divisible by $13$.

Prove that $ 16^{20}+29^{21}+42^{22}$ is divisible by $13$.

This is not a homework question. I would like to know how to solve this type of problems, I solved similar problem with n in exponent, but that could be proved by induction. Here I guess, Euler's theorem could be useful, but I can't work it out. Could anybody give me a hint of how to solve it?

Answer

$16 \equiv 29 \equiv 42 \equiv 3 \text{ mod } 13$. Thus, the claim follows from $3^{20}+3^{21}+3^{22}=3^{20}(1+3+9)=3^{20}13$.