Factor calculator
36 has nine factors: 1, 2, 3, 4, 6, 9, 12, 18 and 36.An odd count means the number is a perfect square, because one factor pairs with itself — 6 × 6 here. Its prime factorisation is 2² × 3², and every factor above is a product of some of those primes.
Every factor of a number, the pairs they come in, and the primes underneath — with the three things the list settles: whether it is prime, whether it is a square, and whether its factors add up to more than itself.
Checked by test against every number to 200 · How we check
9 factors
1, 2, 3, 4, 6, 9, 12, 18, 36
Prime factorisation: 36 = 2² × 3²
The pairs that make it
1 × 36 = 36
2 × 18 = 36
3 × 12 = 36
4 × 9 = 36
6 × 6 = 36
Factors come in pairs, which is why finding them only takes a walk up to the square root — every divisor below it has a partner above. Because 36 is a perfect square, its middle pair is a number multiplied by itself, so the count is odd.
What the list says
| Prime | no |
| Perfect square | yes — 6 × 6 |
| Sum of the factors below it | 55 |
| Classification | abundant — its factors sum to more than itself |
Why the square root is where you stop
Divisors pair off around the square root: for 36, 1 pairs with 36, 2 with 18, 3 with 12, 4 with 9, and 6 with itself. Anything above 6 is the larger half of a pair you have already found. So checking every number up to the root finds all of them, and checking past it finds nothing new — which is why a twelve-digit number factors instantly here rather than after a million divisions.
The same fact explains the odd-count rule. Every factor has a distinct partner except when the number is a perfect square, where the middle pair is a number with itself. That is the whole reason 36 has nine factors and 35 has four, and it holds for every number ever tested.
Questions people actually ask
- What is the difference between factors and prime factors?
- Factors are every number that divides it — 36 has nine of them. Prime factors are the primes it is built from, with their powers: 36 is 2² × 3². The prime factorisation is unique, which is the fundamental theorem of arithmetic, and every one of the nine factors is a product of some subset of it.
- Why do factors come in pairs?
- Because dividing by one gives the other: 36 ÷ 4 = 9, so 4 and 9 are a pair. That is why finding all the factors only takes a walk up to the square root — every divisor below it has a partner above, and beyond the root you would just be listing the same pairs backwards.
- Why do perfect squares have an odd number of factors?
- Because one pair is a number multiplied by itself and so gets counted once instead of twice. 36 has the pair 6 × 6, and 6 is one factor rather than two — which is why 36 has nine factors while 35 has four. Every number with an odd factor count is a perfect square, with no exceptions.
- What is a perfect number?
- One whose factors below itself add up to itself. 6 is the smallest: 1 + 2 + 3 = 6. Then 28, 496 and 8,128 — and only 51 are known, all even, all found by Euclid’s rule from a Mersenne prime. Whether any odd perfect number exists is unsolved after more than two thousand years.
- How do I find the factors by hand?
- Divide by 1, 2, 3 and upward, keeping the pairs, and stop when the divisor passes the square root. For 36 that is 1×36, 2×18, 3×12, 4×9, 6×6 — five divisions and you have all nine factors. Trial division past the root is wasted work, and the pairs column above is that method laid out.