Perfect, Abundant and Deficient Numbers: Add the Proper Divisors

Leave out the number itself

A proper positive divisor is a positive factor smaller than the number. Add those divisors and compare their sum with the original integer. Equal means perfect, greater means abundant, and less means deficient. These are mathematical labels, not judgments of quality.

For 6, the proper divisors are 1, 2 and 3. Their sum is 6, so 6 is perfect. Including 6 itself would give 12 and produce the wrong classification.

Compare three complete examples

For 12, the proper divisors are 1, 2, 3, 4 and 6. They add to 16, which exceeds 12, so 12 is abundant. For 8, the proper divisors are 1, 2 and 4. They add to 7, so 8 is deficient.

Twenty-eight is another perfect number: 1 + 2 + 4 + 7 + 14 = 28. Being a perfect square is unrelated: 16 is a square but its proper divisors sum to 15, so it is deficient.

Special cases help check the definition

Every prime is deficient because its only proper positive divisor is 1. One is also deficient: it has no proper positive divisors, and the empty sum is 0. Neither observation makes these numbers less useful.

The reference pages display both the sum of all positive divisors and the sum excluding the number. Use the second value for this classification. Keeping the two totals separate is a simple way to prevent an off-by-one-factor error.

Check the mathematical definitions

For fractions, percentages and worked arithmetic, see OpenStax Prealgebra. For number definitions, see OpenStax on prime and composite numbers and Wolfram MathWorld on digital roots. Worked examples on this page are calculated for this guide.