I am currently working on Project Euler Problem 23 which involves abundant numbers. In short, abundant numbers are numbers that are less than the sum of their proper divisors. For example, 12 has proper divisors 1,2,3,4,6. The sum 1+2+3+4+6=16 > 12, so 12 is abundant.
Though the Wikipedia article for abundant numbers says that every multiple of an abundant number is abundant, I don't understand why this is true.
I think that this Quick question on abundant numbers on MSE is the rigorous version of what I'm after but I don't understand the proof given (if it is correct or complete).
Is there a simpler/less formal way to understand why multiples of abundant numbers are abundant?
$n$ is abundant if the sum of its divisors is greater than $2n$. Think of a list of those divisors. If you multiply $n$ by some number $k$, you can multiply each divisor in the list by $k$ and it will still divide $kn$. The sum of this new list exceeds $2kn$. In fact, you can now add all the divisors of $k$ except $k$ itself, so the sum is even higher.