# 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 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.

# Highest Common Factor (HCF): 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: . 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; Prime Factor Method

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

.

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

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

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

Shortest list is: .

The least common multiple of 15 and 18 is .

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

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

LCM =

With this list of factors, we can identify both 360 and 400 .

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.