Divisor of Integer/Examples/8 divides 3^2n + 7/Proof 2

Theorem

Let $n$ be an integer such that $n \ge 1$.

Then:

$8 \divides 3^{2 n} + 7$

Proof

From Integer Less One divides Power Less One, we have that:

$\forall m, n \in \Z: \paren {m - 1} \divides \paren {m^n - 1}$

Hence we have the special case where $m = 3^2$:

$8 \divides 3^{2 n} - 1$

from which it follows immediately that:

$8 \divides 3^{2 n} - 1 + 8 = 3^{2 n} + 7$

$\blacksquare$