Factors and Multiples, and How to Find Every Factor

The two words get swapped constantly, and it matters more than it looks, because half of all fraction work depends on knowing which one you need.

Listing every factor without missing one

Work in pairs from 1 upwards. For 36 that is 1 x 36, 2 x 18, 3 x 12, 4 x 9 and 6 x 6. Stop when the pairs meet, which happens at the square root. The list reads 1, 2, 3, 4, 6, 9, 12, 18, 36 (nine factors), and pairing guarantees you did not skip any.

There is a shortcut for the count alone. Prime factorise first: 36 is 2² x 3². Add one to each exponent and multiply, so 3 x 3 is 9. It works for anything. 60 is 2² x 3 x 5, so (2+1)(1+1)(1+1) gives 12 factors, and counting them by hand confirms it.

Factors of a number against multiples of it

Factors of 12 stop at 12. Multiples of 12 start at 12 and never stop. A number is always both its own largest factor and its own smallest multiple, which is the bit that trips people up.

When you need a common denominator you want a common multiple. When you need to reduce a fraction you want a common factor. Reaching for the wrong one is the reason 84/120 sits there refusing to simplify.

GCF, LCM, and one worked pair

Prime factorise both numbers: 84 is 2² x 3 x 7, and 120 is 2³ x 3 x 5. For the greatest common factor, take the lowest power of each shared prime, so 2² x 3 is 12. Divide through and 84/120 becomes 7/10.

For the lowest common multiple, take the highest power of every prime that appears anywhere: 2³ x 3 x 5 x 7 is 840. A useful check falls out of this. GCF times LCM equals the product of the two numbers, and 12 x 840 is 10,080, which is exactly 84 x 120. That identity holds for any pair, so it is a free way to catch a slip.

The same pair shows what the LCM is for. Adding 1/84 and 1/120 needs 840 underneath, so the fractions become 10/840 and 7/840, which add to 17/840. Multiplying the denominators instead gives 10,080, and the answer arrives as 204/10,080, correct but waiting to be simplified back to 17/840.

Simplifying fractions is this same process with the division already done for you.

How to use the factor calculator

Type in a whole number and it returns the full factor list, usually with the prime factorisation beside it. Enter two numbers and you get the greatest common factor and lowest common multiple together. Decimals and negatives are not what it is built for, so multiply up to a whole number first.

Large inputs are where it earns its place. Factoring 4,199 by hand means testing every prime up to 64 and hoping you stay awake; the calculator returns 13 x 17 x 19 straight away. If you are heading towards adding or reducing fractions next, the fraction calculator guide picks up where this leaves off.

Common questions

Is 1 a factor of every number? Yes, and every number is a factor of itself. Those two always sit at the ends of the list, which is why 7 has exactly two factors and counts as prime.

How do I check whether a number is prime? Test divisibility by the primes up to its square root. For 97 that means 2, 3, 5 and 7. None of them divide it, so 97 is prime. You never need to test past the root, because any larger factor would have a smaller partner you already tried.

What are the factors of a prime number? Just 1 and the number itself. That is also why 1 is not prime: it has one factor rather than two, which is a definition rather than an accident.

Do negative numbers have factors? Technically yes, in mirrored pairs, since -12 divides evenly by both 3 and -3. School work and most calculators stick to the positive factors and leave it there.