Question

How do you find the greatest common factor (GCF) of two numbers?

Answer 1

Here are four methods...

Explanation:

There are several methods. Which one is most convenient probably depends on the nature of the numbers.

Method 1 - Factor both numbers first

This method involves expressing each of the numbers as a product of prime numbers and identifying the common factors.

For example:

`252 = 2 * 2 * 3 * 3 * 7`

`90 = 2 * 3 * 3 * 5`

So the GCF of `252` and `90` is:

`2 * 3 * 3 = 18`

Method 2 - Integer division

Given two numbers, proceed as follows:

• Divide the larger number by the smaller to give a quotient and remainder.

• If the remainder is `0` then the smaller number is the GCF.

• Otherwise repeat with the smaller number and the remainder.

So in our example...

`252 / 90 = 2" "` with remainder `72`

`90 / 72 = 1" "` with remainder `18`

`72 / 18 = 4" "` with remainder `0`

So the GCF of `252` and `90` is `18`

Method 3 - Subtraction

Given two numbers, subtract the larger from the smaller until the two numbers are the same, which is the GCF.

So with `252` and `90` it goes like this:

`252 - 90 = 162`

`162 - 90 = 72`

`90 - 72 = 18`

`72 - 18 = 54`

`54 - 18 = 36`

`36 - 18 = 18`

So the GCF of `252` and `90` is `18`

Method 4 - Simultaneous factoring

In this method we identify common factors and discard unique factors until one of the numbers becomes `1` or the numbers are identifiably coprime.

For example, given `252` and `90`...

• Note that both are divisible by `2` since they both end in an even digit. So note down `color(red)(2)` and divide both numbers by `2` to give: `126` and `45`.

• Note that `126` is divisible by `2` but `45` is not. So discard the unique factor `2` by dividing `126` by `2` to leave us: `63` and `45`.

• Note that both `63` and `45` are divisible by `3` since the sum of their digits is divisible by `3`. So note down `color(red)(3)` and divide both numbers by `3` to get: `21` and `15`.

• Note that both `21` and `15` are divisible by `3` again. So note down `color(red)(3)` and divide to get: `7` and `5`.

• `7` and `5` are both prime and have no common factor, so we can stop and gather the factors we found:

  `color(red)(2) * color(red)(3) * color(red)(3) = color(blue)(18)`