Divisor of Integer/Examples/80 divides 9^2n - 1/Proof 2

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Theorem

Let $n \in \Z_{\ge 0}$ be a non-negative integer.

Then:

$80 \divides 9^{2 n} - 1$

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 $m = 9^2$.

$\blacksquare$