# 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 which is 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.

# Practice finding HCF and LCM of small integers

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.

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

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

Common to both:

That is, HCF of 360 and 400 is .

Notice that ; . The numbers and are known as coprime – they don’t have any common factors.

# Lowest Common Multiple Formula

Find the LCM of 15 and 18.

Now,

Note that:

Therefore,

Let’s divide both sides by 3, the highest common factor:

We can be sure that is a multiple of both and (it is and ). We can also be sure that it is the lowest common multiple as 5 and 6 do not share any factor other than 1.