Burt Kaliski Jr. (@modulomathy)
4/21/12 3:27 PM
#math trivia for #April21: #112 is four times a perfect number. Does that make it more or less abundant? Is there a general rule?
The divisors of 112 are 1, 2, 4, 7, 8, 14, 16, 28, 56, and 112. The sum of the divisors less than the number is 136, which makes 112 abundant.
In general, a multiple of a perfect number will be abundant. Let N be a perfect number and let k*N be a multiple, k > 1. Let d1, d2, …, dm denote the divisors of N less than N. By definition, their sum is N.
Observe that the values k*d1, k*d2, …, k*dm are all divisors of k*N, that they are less than kN, and that their sum is k*N. Also observe that k*N has at least one other divisor that is not among these values, 1. Therefore, the sum of the divisors of k*N less than k*N must be at least k*N+1, which implies that k*N is abundant.