HCF and LCM

Highest Common Factor (HCF): simple values

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. For large numbers, it is better to use the prime factor method.

Lowest Common Multiple (LCM) Simple Values

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

Of course, the number 270 which is 15 \times 18 is a common multiple of 15 and of 18. However, 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.

Skill 2a

Highest Common Factor (HCF): the prime factor method

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

24=2 \times 2 \times2 \times 3

30=2 \times 3 \times 5

Prime factors common to both 24 and 30 are: 2\times 3. Therefore 6 is the HCF of 24 and 30.

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

360=2^3 \times 3^2 \times 5 =2 \times 2 \times 2 \times 3 \times 3 \times 5

400 = 2^4 \times 5^2 = 2 \times 2\times 2 \times 2 \times 5 \times 5

Common to both: 2, 2, 2, 5

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

Notice that 360 = 40 \times 9; 400 = 40 \times 10. The numbers 9 and 10 are known as coprime – they don’t have any common factors.

Lowest Common Multiple; Prime Factor Method

Example 1: Let’s use prime factors to find the lowest common multiple of 15 and 18.

15=3\times 5

18 = 2\times 3 \times 3.

Now, our answer needs to be a multiple of 15, so it must have prime factors 3 and 5.

It is also a multiple of 18, so it must also have prime factors 2, 3 and 3.

Let’s write the shortest list of prime factors that includes everything needed for both 15 and 18:

Shortest list is: 3, 5, 2, 3.

The least common multiple of 15 and 18 is 3 \times 5 \times 2 \times 3 = 90.

Example 2: Let’s find the lowest common  multiple of 360 and of 400

360 = 2^3 \times 3^2 \times 5

400 = 2^4 \times 5^2

Let’s begin by using the prime factors of one value, and multiply by the factors from the second list not already present:

LCM = (2^3 \times 3^2 \times 5) \times (2 \times 5)=2^4 \times 3^2 \times 5^2=16 \times 9 \times 25 = 3600

With this list of factors, we can identify both 360 (2^3 \times 3^2 \times 5) and 400 (2^4\times  5^2).

Here are 5 Methods (for a visual method, go to the last page) to help you get to the HCF and the LCM correctly for large values.

Skill 2b

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