HCF and LCM

Use number knowledge to find HCF and LCM of small integers. Use prime factors to calculate HCF and LCM of larger integers.

Highest Common Factor of small integers

Let’s list all the factors of 12 and of 8:

Factors of 12 are: 1, 2, 3, 4 , 6, 12

Factors of 8 are: 1, 2, 4 , 8

By examining the lists, the highest factor common to both integers 4.

This kind of search is known as an exhaustive search – we list all the factors so all possible values are considered. This is a good method for small numbers.

Lowest Common Multiple of small integers

Let’s find the lowest common multiple of 15 and 18.

15 \times 18 =270, so 270 is a common multiple of 15 and of 18. But it is not the lowest common multiple.

Multiples of 15 are: 15, 30, 45, 60, 75, 90 , 105, 120, …

Multiples of 18 are: 18, 36, 54, 72, 90 , good, we can stop here.

The first multiple that we come to that is common to both is 90.

This method is good when the numbers are relatively small.

Practice finding HCF and LCM of small integers

Practice finding the HCF and the LCM by recalling multiplication tables.

Geogebra link

Highest Common Factor using the prime factor method

Example 1: Let’s find the HCF of 24 and 30 using the prime factor method.

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

30=2\times 3 \times 5

We see 2 and 3 in both lists. Therefore 2\times 3=6 is the HCF of 24 and 30.

Example 2: Let’s see how this works with much larger values:

    \begin{align*}360&=2^3 \times 3^2 \times 5 &=2 \times 2 \times 2 \times 3 \times 3 \times 5\\[12pt]400 &= 2^4 \times 5^2 &= 2 \times 2\times 2 \times 2 \times 5 \times 5\end{align}

We see 2, 2, 2, 5 in both lists.

That is, HCF of 360 and 400 is 2\times 2 \times 2 \times 5 = 40.

Notice that

    \begin{align*}360 &= 40 \times 9\\[10pt]400 &= 40 \times 10\end{align}

The numbers 9 and 10 are known as coprime – they don’t have any common factors.

Here is a visual:

GeoGebra link

Lowest Common Multiple Formula

Find the LCM of 15 and 18.

Notice, HCF(15, 18)=3

Here is a logical argument. Find the shortest list of factors that has all the factors of 18 and all the factors of 15:

First, Take the largest number and examine the factors. 18 requires 2, 3, 3

Next, examine the factors of 15: we need 3 and 5.

18 has a 3 but 18 does not have the factor 5.

The shortest list that has all the factors of both numbers is 2, 3, 3, 5.

Therefore, LCM(18,15) = 2 \times 3 \times 3 \times 5 = 90.

Alternatively, we can use this formula:

    \[LCM(m,n)=\frac{m \times n}{HCF(m,n)}\]

Therefore

    \[LCM(15,18)=\frac{15 \times 18}{3}=90\]

Practice finding HCF and LCM of larger integers

Use the prime factor method to find the HCF of the two numbers given. Then use the formula or otherwise to find the LCM.

Geogebra link

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