[Math] Prove that $3n^7 + 7n^3 + 11n$ is divisible by $21$ for all integers $n$

Question

Prove that $3n^7 + 7n^3 + 11n$ is divisible by $21$ for all integers $n$

I needed some help solving this. I know that we must show that it is divisible by 3 and 7 but how do I show that

$$ 3n^7 + 7n^3 + 11n \equiv 0 \mod{3} $$

Answer 1

Note that $n^3\equiv n\mod 3$ since $n^2\equiv 1\text{ or } 0\mod 3$. Thus the equation becomes $$0+n+2n \equiv 0\mod 3$$ which is clearly true. For $7$ the strategy is similar, since $n^7\equiv n\mod 7$ so the equation becomes $3n+0+4n\equiv 0\mod 7$ which is again clearly true.