I arrived at the solution. It was much simpler than what I attempted in the original post.
I need to prove that a multiple of a perfect number is abundant.A positive integer n is abundant if sigma(n) > 2n (where sigma(n) = sum of divisors of n)
An integer k is prefect if sigma(k) = 2k
Suppose n is a prefect number and k is a nonnegative integer.
If gcd(n,k)=1 then sigma(nk) = sigma(n) * sigma(k) = 2n * sigma(k) > 2nk
Otherwise... this is where I'm having some trouble.. any suggestions?