## Even Sum Of Two Abundant Numbers

#### Problem

The divisors of a positive integer, excluding the number itself, are called the proper divisors . If the sum of proper divisors is equal to the number we call the number perfect. For example, the divisors of $28$ are $1, 2, 4, 7, 14,$ and $28,$ so the sum of proper divisors is $1 + 2 + 4 + 7 + 14 = 28$.

Similarly, if the sum of the proper divisors exceeds the number we call the number abundant. For example, $12$ is abundant because the divisors of $12$ are $1, 2, 3, 4, 6, 12,$ and the sum of proper divisors, $1 + 2 + 3 + 4 + 6 = 14 \gt 12$.

Prove that every even $n \ge 48$ can be written as the sum of two abundant numbers.

Problem ID: 347 (08 Nov 2008) Difficulty: 4 Star