Divisors Calculator
| n | Proper divisors | Sum minus n | Verdict |
|---|---|---|---|
| 6 | 1, 2, 3 | 6 | perfect |
| 28 | 1, 2, 4, 7, 14 | 28 | perfect |
| 496 | 1, 2, 4, 8, 16, 31, 62, 124, 248 | 496 | perfect |
| 12 | 1, 2, 3, 4, 6 | 16 | abundant |
| 15 | 1, 3, 5 | 9 | deficient |
| 220 | 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 | 284 | abundant, amicable with 284 |
A divisor of n is a whole number that divides it with nothing left over, and the full list hides more structure than it seems: the Greeks noticed 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14 - numbers equal to the sum of their proper divisors, which they named perfect. This calculator lists every divisor of any number up to a trillion, counts them, sums them, and files the number as deficient, abundant or perfect.
The three verdicts split the integers: deficient numbers (like 15, whose proper divisors sum to 9) are most of them; abundant numbers (like 12, whose divisors oversum to 16) waste a little; perfect numbers sit exactly on the line and are so rare that after 8128 the next one is 33,550,336. Nobody has ever found an odd perfect number - if one exists it is above 10¹⁵⁰⁰, and finding it would be a career.
The connections run deep: divisor counts come from prime factorization by the (exponent+1) rule, even perfect numbers are locked to Mersenne primes by the Euclid-Euler theorem, and amicable pairs like 220 and 284 - each the other's proper-divisor sum - were a friendship charm from Pythagoras through medieval letters. All of that falls out of one sorted list.
How to use
- Type a whole number up to a trillion - the list pairs divisors (i with n/i) so it only checks up to the square root.
- Read the count and sum - the count comes from the factorization exponents, the sum is what the verdict is built on.
- Check the verdict: sum of proper divisors less than n means deficient, more means abundant, exactly equal means perfect.
- Try 220 and then 284 - each one's proper divisors sum to the other, the amicable pair ancient mathematicians carved into love charms.
Frequently asked questions
What are the divisors of a number?
Every whole number that divides it exactly: 36 has divisors 1, 2, 3, 4, 6, 9, 12, 18, 36 - nine of them. Divisors come in pairs (i and n/i), which is why a calculator only needs to test up to the square root, and why perfect squares like 36 have an odd divisor count.
What is a perfect number?
A number equal to the sum of its proper divisors: 6 = 1 + 2 + 3, 28 = 1 + 2 + 4 + 7 + 14, then 496 and 8,128. The first four were known to ancient Greece; every even perfect number has the Euclid-Euler form 2^(p-1) x (2^p - 1) with the second factor a Mersenne prime - and no odd perfect number has ever been found.
What are abundant and deficient numbers?
Compare the sum of proper divisors to the number itself: 12's divisors 1+2+3+4+6 = 16 overshoot, so 12 is abundant; 15's sum to 9 undershoot, so it is deficient. Nearly every number is deficient; abundant numbers start at 12 and are always multiples of relationship-rich factorizations.
What is an amicable pair like 220 and 284?
Two numbers whose proper divisors sum to each other: 220's divisors (1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110) add to 284, and 284's (1, 2, 4, 71, 142) add back to 220. Pythagoreans called the pair a symbol of friendship, and medieval love charms traded on exactly this arithmetic.
How is the divisor count related to prime factorization?
By the (exponent+1) rule: 360 = 2³ × 3² × 5 has (3+1)(2+1)(1+1) = 24 divisors - each divisor chooses an exponent for every prime from 0 up to its exponent. Listing them confirms it; the factorization predicts it. That link is why the factorization calculator and this page are two views of the same theorem.