INTRODUCING the idea of the greatest 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.

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.

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.