Finding the Highest Common Factor (HCF) of Two Numbers

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The highest common factor is the largest number that divides two number. For example the highest common factor of 6 and 8 is 2 because 2 divides 6 and 2 divides 8 but there is no larger number that divides both 6 and 8. Alternatively we could write down all the factors of 6 in a list, all the factors of 8 in a list and pick the biggest number in both lists:

Factors of 6: 1, 2, 3, 6

Factors of 8: 1, 2, 4, 8

The largest number in both lists is 2 so 2 is the highest common factor. For larger numbers the list may be long and it is quite easy to miss some factors so we need a better method. The most reliable is the tree method, to write both numbers as products of prime numbers.. This works as follows:

Find the highest common factor of 24 and 36.

Write 36 as a product of two numbers. Any two will do but neither number can be 1. I chose 6 and 6.

Write the 6’s as products of numbers. The only possible way is 3*2 for each.

Example

Write24 as a product of two numbers. Any two numbers will do but neither number can be 1. I chose 4 and 6.

Write the 4 and 6 as products of numbers, neither of which can be 1. The only possible choices are4=2*2 and 6=3*2.

Now 36=3*2*3*2 and 24=2*2*3*2

We have 2 2’s and a 3 in both lists so the highest common factor (HCF) is 2*2*3=12

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