3.1.2 Perfect, Abundant, and Deficient Numbers

For this section let’s begin with the definitions of each type.

A perfect number has the sum of the proper divisors equal to itself. The first three perfect numbers are 6 ($1 + 2 + 3 = 6$), 28 ($1 + 2 + 4 + 7 + 14 = 28$), and 496 ($1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496$). Notice that there are really only two perfect numbers that would be reasonable to test on a number sense test (6 and 28 should be memorized as being perfect).

An abundant number has the sum of the proper divisors greater than itself. Examples of an abundant number is 12 ($1 + 2 + 3 + 4 + 6 = 16 > 12$) and 18 ($1 + 2 + 3 + 6 + 9 = 21 > 18$). An interesting property of abundant numbers is that any multiple of a perfect or abundant number is abundant. Knowing this is very beneficial to the number sense test.

As you can assume through the process of elimination, a deficient number has the sum of the proper divisors less than itself. Examples of these include any prime number (because they have only one proper divisor which is 1), 10 ($1 + 2 + 5 = 8 < 10$), and 14 ($1 + 2 + 7 = 10 < 14$) just to name a few. An interesting property is that any power of a prime is deficient (this is often tested on the number sense test).