Multiple of Abundant Number is Abundant

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Theorem

Let $n$ be an abundant 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:

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

But from Abundancy Index of Product is greater than Abundancy Index of Proper Factors:

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

Hence the result by definition of abundant.

$\blacksquare$


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