Highest Common Factor/Lowest Common Multiple 

Start off from Key Skills

The first step in finding both the HCF (Highest Common Factor) and the LCM (Lowest Common Multiple) is to first decompose the numbers in question into a multiple of their prime factors.

Proceed by example:

Decompose 28 into a multiple of its prime factors

28 = 2 × 14
   = 22 × 7

Decompose 84 into a multiple of its prime factors

84 = 2 × 42
   = 22 × 21
   = 22 × 3 × 7

Decompose 180 into a multiple of its prime factors

180 = 2 × 90
   = 22 × 45
   = 22 × 3 × 15
   = 22 × 32 × 5

Lowest Common Multiple

Find the LCM of 4,10 and 22

4 = 22
10 = 2 × 5
12 = 2 × 6 = 22 × 3

Extract the highest power of each prime factor appearing above, and form a multiple, i.e.

22 × 3 × 5 = 60

You can usually mentally check your answer to see if it is right, or looks right - we are looking for the number that is a multiple of all three original numbers and we don't want there to be any lower number that is also a multiple of all three numbers.

Highest Common Factor

Find the HCF of 10, 15 and 30

10 = 2 × 5
15 = 3 × 5
30 = 2 × 15 = 2 × 3 × 5

This time, we extract the lowest power of prime factors appearing above but only for those prime factors that appear in the decompostion of each of the original numbers. So for the above number, the Highest Common Factor is

5

(5 is the only factor that appears in all decompositions and its lowest power is just 1, corresponding to just 5 itself)

Find the HCF of 28, 42 and 84

28 = 2 × 14 = 22 × 7
42 = 2 × 21 = 2 × 3 × 7
84 = 2 × 42 = 22 × 21 = 22 × 3 × 7

So the Highest Common Factor is

2 × 7 = 14

(3 does not apear in all decompositions so is not considered. The lowest power of 2 is one, i.e. corresponding to just 2 itself)

Quick Quiz    

Find the L.C.M. of
a) 10, 15, 40
b) 8,12
c) 10, 25
d) 4,7,21

Find the HCF of
a) 9,12
b) 16, 35
c) 8, 40, 56
d) 26, 39, 52
e) 42, 98, 112