How many factors?

Hello, and welcome to Myriora Maths! How many factors does seventy-two have? You could list them all, but it's easy to miss one. There's a clever way to count them, using prime factors. Let's see how!

Start small, with twelve. Twelve is two times two times three. Every factor of twelve is made from some of those primes. You can use no twos, one two, or two twos: that's three choices. And no three, or one three: two choices. Three choices times two choices: six factors!

Let's check by listing them: one, two, three, four, six and twelve. Six factors!

So here's the rule. Write the number as prime factors. For each prime, count how many times it appears, and add one. Then multiply those numbers together.

Now seventy-two. It's two times two times two, times three times three: three twos and two threes. For the twos: three plus one, so four choices. For the threes: two plus one, so three choices. Four times three: seventy-two has twelve factors!

Why add one? Because choosing none of a prime is a choice too. Choose none of every prime, and you get the factor one, which uses no primes at all!

Here's a question that catches lots of people. How many factors does seventy-two have? Some people multiply the counts: three twos and two threes, and three times two is six. But they forgot the choice of none! It's four times three: twelve.

Your turn! How many factors does a hundred have? Here's a hint: a hundred is two times two times five times five. Pick the right card. You've got twenty seconds, or pause the video. Go!

Time's up! A hundred has two twos and two fives. The twos give three choices, and the fives give three choices. Three times three: a hundred has nine factors!

Quick review. Break the number into prime factors. For each prime, count how many times it appears, and add one. Multiply those numbers together. And don't forget the plus one: none is a choice too! See you next time!