Factor trees

Hello, and welcome to Myriora Maths! Every whole number bigger than one is built out of prime numbers. Today, we'll grow a factor tree to find them. Let's go!

Why would we want to break a number into primes? Imagine you have sixty sweets to share fairly. Can you share them between four friends? Five? Seven? Sixty is two, times two, times three, times five. So it shares fairly by any number you can build from those primes: two, three, four, five, six, ten, and more. But not seven! And later, primes make fractions much easier too.

First, a quick reminder. A prime number has exactly two factors: one, and itself. Two, three, five, seven, eleven. You can't split them any further. They are the building blocks of every other number.

Let's start with sixty. Split it into any two numbers that multiply to make sixty. Let's use six and ten. Six isn't prime, so it splits again: two times three. Ten splits too: two times five. Now every branch ends in a prime number. Circle them!

So, sixty equals two, times two, times three, times five. We can also write it with a power. Two times two is two squared, so sixty is two squared, times three, times five.

Now here's the amazing part. What if we had started differently? Sixty is also four times fifteen. Four splits into two times two. Fifteen splits into three times five. Look! The same primes. Every number has only one set of prime factors, however you grow the tree.

Two tips for growing trees. One: if you get stuck, try dividing by the smallest primes first. Two, then three, then five. Two: keep going until every branch ends in a prime.

Let's try one together. Grow a factor tree for thirty-six. Split it into any two numbers that multiply to make thirty-six. Say your split out loud!

You could start with six times six. Each six splits into two times three. So thirty-six is two, times two, times three, times three. Whatever you start with, you get the same primes!

Here's a question that catches lots of people. Is this factor tree finished? Thirty-six splits into four and nine, and four splits into two times two. Some people stop there, because nine looks a bit like a prime. But nine is three times three, so its branch still has to split.

Your turn! Which card shows eighty-four as a product of primes? You've got fifteen seconds, or pause the video. Go!

Time's up! Two times forty-two isn't finished, because forty-two isn't prime. Four times three times seven isn't finished either: four is two times two. So it's two, times two, times three, times seven!

Quick review. Prime numbers are the building blocks of every number. A factor tree splits a number until every branch is prime. And however you grow the tree, you always get the same primes. See you next time!