Introduction
When we compare two numbers, we often want to know what their highest common factor and lowest common multiple are. These numbers help us understand how they are related, and are often the answer to problems involving them.
Highest common factor
A number is a common factor of two other numbers if it divides both numbers without remainder. Below is a table with all the factors of 24 and 54, with their common factors listed in the last row:
| \text{Factors of }24 | 1,\,2,\,3,\,4,\,6,\,8,\,12,\,24 |
|---|---|
| \text{Factors of }54 | 1,\,2,\,3,\,6,\,9,\,18,\,27,\,54 |
| \text{Common factors} | 1,\,2,\,3,\,6 |
The common factors tell us how we can break up both numbers equally.
For example, 24 students and 54 books can be divided into:
2 equal groups with 12 students and 27 books each
3 equal groups with 8 students and 18 books each
6 equal groups with 4 students and 9 books each
The largest number in the list of common factors, 6, is called the highest common factor (HCF) of 24 and 54.
To find the HCF of two numbers we can always create a list of factors, like we did above. But there is a faster way using factor trees. To find the highest common factor of 126 and 294, we start by drawing their factor trees:
We then find what factors appear in both factorisations - they have one 2, one 3, and one 7 in common. The highest common factor is the product of the common prime factors, 2 \times 3 \times 7 = 42.
Exploration
Use the following applet to explore highest common factors. Select any two numbers and use the slider on the right of each number to reveal the factors of that number.
The highest common factor is the largest common factor in the lists of factors.
The highest common factor (HCF) of two numbers is the largest factor of both numbers. It is the product of the common prime factors between them.
Examples
Example 1
Find the highest common factor of 150 and 880.
The highest common factor (HCF) of two numbers is the largest factor of both numbers. It is the product of the common prime factors between them.
Lowest common multiple
When we multiply two numbers together, these two numbers are always factors of the new number. Another way to look at it is to say the new number is a common multiple of both of the original ones.
Below is a table with the first few multiples of 12 and 18, with their common multiples listed in the last row:
| \text{Multiples of }12 | 12, 24, 36, 48, 60, 72, 84, 96, 108, \ldots |
|---|---|
| \text{Multiples of }18 | 18, 36, 54, 72, 90, 108, 126, \ldots |
| \text{Common multiples} | 36, 72, 108, \ldots |
The common multiples are the numbers that can be broken up into both the original numbers equally.
- 36 can be broken up into 3 groups of 12, and 2 groups of 18
- 72 can be broken up into 6 groups of 12, and 4 groups of 18
- 108 can be broken up into 9 groups of 12, and 6 groups of 18
The smallest number in the list of common multiples, 36, is called the lowest common multiple (LCM) of 12 and 18.
To find the LCM of two numbers we can always create a list of multiples, like we did above. But just like for HCF, there is a faster way using factor trees. To find the lowest common multiple of 126 and 294, we look again at their factor trees and jump straight to the prime factorisations in expanded form, since we found that already:
Now we create the number by multiplying one of the numbers by any prime factors it is missing from the other one.
There is only one 2 in each, there are two 3s in 126 (and only one in 294), but there are two 7s in 294.
If we multiply 126 by 7, the missing factor, we will find the least common multiple: 126 \times 7 = 882.
We would get the same result if we started with 294 - it has more than enough 7s, but it is missing the second 3 that 126 has. The multiplication 294 \times 3 = 882 gives us the same answer.
The lowest common multiple (LCM) of two numbers is the smallest multiple of both numbers. Multiply one number by the prime factors it is missing from the other.
Examples
Example 2
Consider the following prime factorisations: \begin{aligned} 2200 &= 2 \times 2 \times 2 \times 5 \times 5 \times 11 \\2750 &= 2 \times 5 \times 5 \times 5 \times 11 \end{aligned}
Find the lowest common multiple of 2200 and 2750.
The lowest common multiple (LCM) of two numbers is the smallest multiple of both numbers. Multiply one number by the prime factors it is missing from the other.