Congruence of Powers

Theorem
Let $a, b \in \R$ and $m \in \Z$.

Let $a$ be congruent to $b$ modulo $m$, that is:
 * $a \equiv b \pmod m$

Then:
 * $\forall n \in \Z_{\ge 0}: a^n \equiv b^n \pmod m$

Proof
Proof by induction:

For all $n \in \Z_{\ge 0}$, let $P \paren n$ be the proposition:
 * $a \equiv b \implies a^k \equiv b^k \pmod m$

$P(0)$ is trivially true, as $a^0 = b^0 = 1$.

$P(1)$ is true, as this just says:
 * $a \equiv b \pmod m$

Basis for the Induction
$P(2)$ is the case:
 * $a^2 \equiv b^2 \pmod m$

which follows directly from the fact that Modulo Multiplication is Well-Defined.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $P \paren k$ is true, where $k \ge 2$, then it logically follows that $P \paren {k + 1}$ is true.

So this is our induction hypothesis:


 * $a \equiv b \implies a^k \equiv b^k \pmod m$.

Then we need to show:


 * $a \equiv b \implies a^{k + 1} \equiv b^{k + 1} \pmod m$.

Induction Step
This is our induction step:

Suppose $a^k \equiv b^k \pmod m$.

Then $a^k a \equiv b^k b \pmod m$ by definition of modulo multiplication.

Thus $a^{k + 1} \equiv b^{k + 1} \pmod m$.

So $P \left({k}\right) \implies P \left({k + 1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\forall n \in \Z_+: a \equiv b \implies a^n \equiv b^n \pmod m$.