Myriora Maths › Maths foundations › Chapter 6: Factors, multiples and primes

Highest common factor

Ages 9–11Chapter 6 · Lesson 8

In this lesson

The H C F is the biggest number that divides both.
You can list the factors, and pick the highest one in both lists.
Or you can multiply the prime factors they share.
And it's how you simplify fractions.

Your turn

What is the highest common factor of twenty-four and thirty-six?

Show the answer

Twenty-four is two times two times two times three. Thirty-six is two times two times three times three. They share two twos and one three. Two times two times three is twelve!

The common mistake

Here's a question that catches lots of people. What is the H C F of twelve and eighteen? Some people answer two, because two goes into both. But two is just a common factor. The question wants the highest one: six!

Try it at home

Find the biggest equal pieces you can cut two lengths of string or ribbon into, with nothing left over.

Read the whole lesson

Hello, and welcome to Myriora Maths! What's the biggest number that divides both twelve and eighteen? It's called the highest common factor, or H C F. It's how you simplify the fraction twelve eighteenths, and it's the key to lots of puzzles. Let's find it, two ways!

Way one: list the factors. Twelve: one, two, three, four, six, twelve. Eighteen: one, two, three, six, nine, eighteen. The common factors are in both lists: one, two, three and six. The highest is six!

Way two uses prime factors, like the gems in a factor tree. Twelve is two times two times three. Eighteen is two times three times three. Pair up the primes they share: one two, and one three. Multiply the shared primes: two times three is six. Same answer!

Why does this work? A common factor has to be built from primes that both numbers have. Use every prime they share, and you get the biggest one possible.

Prime factors win for bigger numbers. Eighty-four is two times two times three times seven. Ninety is two times three times three times five. They share one two, and one three. So the H C F is two times three: six.

Here's what the H C F is for. Divide the top and bottom of twelve eighteenths by six, and you get two thirds: the simplest it can be. And if you cut a twelve centimetre ribbon and an eighteen centimetre ribbon into equal pieces, as long as possible, each piece is six centimetres.

Your turn! What is the highest common factor of twenty-four and thirty-six? Pick the right card. You've got twenty seconds, or pause the video. Go!

Time's up! Twenty-four is two times two times two times three. Thirty-six is two times two times three times three. They share two twos and one three. Two times two times three is twelve!

Quick review. The H C F is the biggest number that divides both. You can list the factors, and pick the highest one in both lists. Or you can multiply the prime factors they share. And it's how you simplify fractions. See you next time!