The rule
divisors come in pairs multiplying to n
Why it works
Divisors always pair up: if 4 divides 36, so does 9, because 4 × 9 = 36. That pairing means you only ever need to test up to the square root, and each hit gives you two divisors at once.
How to do it by hand
- Test every number from 1 up to the square root
- Each one that divides exactly gives a divisor and its partner
- A perfect square has one unpaired divisor: its root
- Add the proper divisors, meaning all except the number itself
What is worth knowing
The classification by divisor sum goes back to the Pythagoreans and is still unresolved in places. A number whose proper divisors add to itself is perfect: 6 = 1 + 2 + 3. Fewer and it is deficient, more and it is abundant. Every known perfect number is even, and whether an odd one exists has been open for over two thousand years — it is one of the oldest unsolved problems in mathematics. Abundant numbers are common, deficient ones are the majority, and perfect ones are vanishingly rare: only 51 are known.