Divisor of Integer/Examples/63 divides 8^2n - 1/Proof 2

Theorem

Let $n \in \Z_{\ge 0}$ be a positive integer.

Then:

$63 \divides 8^{2 n} - 1$

where $\divides$ denotes divisibility.

Proof

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

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

This result is the special case where $m = 8^2$.

$\blacksquare$