Factors & multiples

Why do 12 cookies split so neatly onto 3 or 4 plates, while 13 fits nowhere evenly? Because some numbers divide cleanly and others don't. That's exactly what this is about: factors and multiples, and how they let you spot at a glance what fits together.

What are factors and multiples?

A factor of 12 is any number that fits into 12 without a remainder: 1, 2, 3, 4, 6 and 12. A multiple of 3 is the reverse: any number you reach by adding 3 again and again: 3, 6, 9, 12 and so on. So factors sit inside a number, multiples build on top of it.

Prime factorisation

Every whole number from 2 upwards can be written as a product of primes, and apart from the order of the factors there is only one way to do it. 12 is $2 \cdot 2 \cdot 3$, 18 is $2 \cdot 3 \cdot 3$. This breakdown is like a number's fingerprint: from it you read off every factor and instantly see what two numbers share.

The greatest common divisor

The greatest common divisor (gcd) of two numbers is the largest number that fits into both without a remainder. For 12 and 18 the common factors are 1, 2, 3 and 6, and the greatest of those is 6. Prime factors make it quick: take each prime as often as it appears in both breakdowns and multiply those factors. For 12 ($2 \cdot 2 \cdot 3$) and 18 ($2 \cdot 3 \cdot 3$) that is one 2 and one 3, so $2 \cdot 3 = 6$.

The least common multiple

The least common multiple (lcm) is the smallest number that is a multiple of both numbers at once. You find it by listing the multiples of both numbers and seeing where they first meet. For 4 and 6 those are 4, 8, 12 and 6, 12, so the lcm is 12. Gcd and lcm are closely linked: their product always equals the product of the two starting numbers.