The Highest Common Factor

The highest common factor (HCF) of two or more numbers is the largest number that is a common factor of the numbers. To find it, write down the numbers as their prime factors. Then choose those factors that appear in all of the numbers.

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Let’s look at some examples.

Examples

Example 1:

Find the HCF of the numbers 15, 24 and 30.

Solution

$15=3\times 5$

$24=2\times 2\times 2\times 3$

$30=2\times 3\times 5$

We see that 3 is common in them all. So the HCF is 3.

Example 2:

Find the HCF of the numbers 16, 24, 72 and 80.

Solution

$16=2\times 2\times 2\times 2$

$24=2\times 2\times 2\times 3$

$72=2\times 2\times 2\times 3\times 3$

$80=2\times 2\times 2\times 5$

We see that $2\times 2\times 2$ appears in them all, so the HCF is $8$ .

Example 3:

Find the HCF of 1245 and 3000.

Solution

To find the prime factors, we use the method of long division.

$3|1245$

$5|415$

$83|83$

$1$

And

$2|3000$

$2|1500$

$2|750$

$3|375$

$5|125$

$5|25$

$5|5$

$1$

So, we have $1245=3\times 5\times 83$ and $3000=2\times 2\times 2\times 3\times 5\times 5\times 5$

$3\times 5$ is common in both, therefore HCF is $3\times 5=15$

Example 4

Find the HCF of 210, 315, and 650.

Solution

We first find the prime factors of each number:

$210 = 2 \times 3 \times 5 \times 7$

$315 = 3^2 \times 5 \times 7$

$650 = 2 \times 5^2 \times 13$

The HCF of 210, 315, and 650 is 5.

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