Unit 2 Section 4 : Highest Common Factor and Lowest Common Multiple

This section introduces the idea of the highest common factor and lowest common multiple of a pair of numbers.

The highest common factor (HCF) of two whole numbers
is the largest whole number which is a factor of both.

HCF Example

Consider the numbers 12 and 15:
The factors of 12 are : 1, 2, 3, 4, 6, 12.
The factors of 15 are : 1, 3, 5, 15.
1 and 3 are the only common factors (numbers which are factors of both 12 and 15).
Therefore, the highest common factor of 12 and 15 is 3.

Example Questions
Work out the answers to these questions and click the buttons marked to see whether you are correct.

(a) Find the highest common factor of 20 and 30 by following the steps below:

What are the factors of 20?

What are the factors of 30?

What is the highest common factor of 20 and 30?

(b) Find the highest common factor of 14 and 12 by following the steps below:

What are the factors of 14?

What are the factors of 12?

What is the highest common factor of 14 and 12?

The lowest common multiple (LCM) of two whole numbers
is the smallest whole number which is a multiple of both.

LCM Example

Consider the numbers 12 and 15 again:
The multiples of 12 are : 12, 24, 36, 48, 60, 72, 84, ....
The multiples of 15 are : 15, 30, 45, 60, 75, 90, ....
60 is a common multiple (a multiple of both 12 and 15), and there are no lower common multiples.
Therefore, the lowest common multiple of 12 and 15 is 60.

Example Questions
Work out the answers to these questions and click the buttons marked to see whether you are correct.

(a) Find the lowest common multiple of 5 and 7 by following the steps below:

What are the first ten multiples of 5?

What are the first ten multiples of 7?

What is the lowest common multiple of 5 and 7?

(b) Find the lowest common multiple of 6 and 10 by following the steps below:

What are the first ten multiples of 6?

What are the first ten multiples of 10?

What is the lowest common multiple of 6 and 10?

Although the methods above work well for small numbers, they are
more difficult to follow with bigger numbers. Another way to find the
highest common factor and lowest common multiple of a pair of
numbers is to use the prime factorisations of the two numbers.

Finding HCF & LCM with prime factorisations

We want to find the HCF and LCM of the numbers 60 and 72.

Start by writing each number as a product of its prime factors.

60 = 2 * 2 * 3 * 5

72 = 2 * 2 * 2 * 3 * 3

To make the next stage easier, we need to write these so that each new prime factor begins in the same place:

60 = 2 * 2 * 3 * 5

72 = 2 * 2 * 2 * 3 * 3

All the "2"s are now above each other, as are the "3"s etc. This allows us to match up the prime factors.

The highest common factor is found by multiplying all the factors which appear in both lists:

So the HCF of 60 and 72 is 2 × 2 × 3 which is 12.

The lowest common multiple is found by multiplying all the factors which appear in either list:

So the LCM of 60 and 72 is 2 × 2 × 2 × 3 × 3 × 5 which is 360.

Example Questions

(a) Find the highest common factor and lowest common multiple of 50 and 70 by following the steps below:
What is 50 as a product of its prime factors?

What is 70 as a product of its prime factors?

What is the HCF of 50 and 70? (find it by multiplying together the factors which appear in both lists above)

What is the LCM of 50 and 70? (find it by multiplying together the factors which appear in either list above)

(b) Find the highest common factor and lowest common multiple of 900 and 270 by following the steps below:
What is 900 as a product of its prime factors?

What is 270 as a product of its prime factors?

What is the HCF of 900 and 270? (find it by multiplying together the factors which appear in both lists above)

What is the LCM of 900 and 270? (find it by multiplying together the factors which appear in either list above)

Exercises

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You have now completed Unit 2 Section 4

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(a) 6 and 9 :
(b) 14 and 18 :
(c) 30 and 24 :
(d) 15 and 10 :

(a) 90 and 120 :
(b) 96 and 72 :
(c) 56 and 60 :
(d) 77 and 50 :
(e) 300 and 550 :
(f) 320 and 128 :

(a) 5 and 3 :
(b) 9 and 6 :
(c) 8 and 10 :
(d) 12 and 9 :
(e) 15 and 20 :
(f) 6 and 11 :

(a) 28 and 30 :
(b) 16 and 24 :
(c) 20 and 25 :
(d) 60 and 50 :
(e) 12 and 18 :
(f) 21 and 35 :