<s>[INST] Show that $1+2^n+2^{2n}$ is divisible by $7$, when $n$ is not a multiple of $3$ Problem taken from a paper on mathematical induction by Gerardo Con Diaz. Although it doesn't look like anything special, I have spent a considerable amount of time trying to crack this, with no luck.
Show that $1+2^n+2^{2n}$ is divisible by $7$, when $n$ is not a multiple of 3.
 [/INST] $n=3k+1$, $1+2^{3k+1}+2^{2(3k+1)}$, $2^3=1$ mod $7$ implies that
$1+2^{3k+1}+2^{2(3k+1)}=1+2+4$ mod $7$.
$n=3k+2$, $1+2^{3k+2}+2^{2(3k+2)}=1+4+16$ mod $7$ and $1+4+16=0$ mod $7$.
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