What is the definition of "perfect number"? What is the definition of "abundant number"? If it is not clear, try some simple examples. The smallest perfect number is 6. It is "perfect" because its proper divisors are 1, 2, and 3 and those add to 6. Okay, multiply 6 by p. 1, 2, and 3 are still proper divisors. So are p, 2p and 3p. Those three alone add to 6p!