Multiple of Perfect Number is Abundant

Theorem

Let $n$ be a perfect number.

Let $m$ be a positive integer such that $m > 1$.

Then $m n$ is abundant.

Proof

We have by definition of $\sigma$ function and perfect number that:

$\dfrac {\map \sigma n} n = 2$
$\dfrac {\map \sigma {m n} } {m n} > 2$

Hence the result by definition of abundant.

$\blacksquare$