HCF and LCM

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. For small numbers, this is a fast approach especially if it can be done mentally. For large numbers, it can be better to use the prime factor method.

Lowest Common Multiple of small integers

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

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

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 \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 Formula

Find the LCM of 15 and 18.

Now, HCF(15, 18)=3

Note that:

    \[18 = 3 \times 6\]

    \[15 = 3 \times 5\]

Therefore,

    \[18 \times 15 = 3 \times 6 \times 3 \times 5\]

Notice that 3, the HCF, is repeated when we multiply 18 \times 15.

Because we used 3, the HCF, the divisors 5 and 6 share no common factor.

We can be sure that 6 \times 3 \times 5 is a common multiple (it is 6 \times 15 and 5 \times 18). We can also be sure that it is the lowest common multiple as 5 and 6 do not share any factor that could be removed.

This leads to the formula:

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

Therefore

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

Skill 2b

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

 


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