Divisibility of fourths by seven

Question

I am otherwise very good in mathematics, but recently I came upon a problem that I just can't solve. Do you have any idea how to solve it?

If $a^2 + b^2 + c^2$ is divisible by 7, prove that $a^4 + b^4 + c^4$ is divisible by 7 as well.

Thanks!

Answer 1

Modulo $7$, squares are congruent to $0$, $1$, $2$ or $4$. The only way a trio of these can add to zero modulo $7$ is for all of them to be $0$, or for them to be some permutation of $1$, $2$ and $4$. In the latter case, $a^4+b^4+c^4 \equiv 1^2+2^2+4^2\pmod 7$.