## Show that one Gaussian integer divides another

Prove that $-1+2i$ divides $7^16-1$ in $\mathbb{Z}[i]$.

First, we claim that $5|(7^16-1)$. To see this, note that $7^{16}-1 \equiv 2^{16}-1 \equiv (2^4)^4-1 \mod 5$. By Fermat’s Little Theorem, $2^4 \equiv 1 \mod 5$. Thus $7^{16}-1 \equiv 0 \mod 5$ as desired. Note also that $-1+2i$ divides 5 in $\mathbb{Z}[i]$; in particular, $(-1+2i)(-1-2i) = 5$. Since divisibility is transitive, $-1+2i$ divides $7^{16}-1$.

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