Finding LCM and HCF from prime factors
Sometimes you are given numbers expressed as a product of prime factors. For example, 8 = 23 and 90 = 2 × 32 × 5.
If you want to find the LCM and HCF in an exam, we can use prime factor form to simplify the process.
Example one
Find the LCM and HCF of 18 and 30.
Solution
Firstly, we write the numbers as a product of prime factors.
18 = 2 × 3 × 3 = 2 × 32
30 = 2 × 3 × 5
Then we create a Venn diagram for the factors:
Once we have the Venn diagram, finding the LCM is simply a matter of multiplying all the numbers in the Venn diagram together:
LCM = 3 × 2 × 3 × 5 = 90
To find the HCF we multiply the numbers in the overlapping quadrant together:
HCF = 2 × 3 = 6
Example two
Find the LCM and HCF of 50 and 16.
Solution
Firstly, we write the numbers in prime factor form:
50 = 2 × 5 × 5 = 2 × 52
16 = 2 × 2 × 2 × 2 = 24
We then draw the Venn diagram:
As both numbers have 2 as a factor, this goes in the middle. The remaining factors go in their respective circles.
The LCM is found by multiplying all of the numbers in the Venn diagram together. As there are four 2s and two 5s:
LCM = 2 × 2 × 2 × 2 × 5 × 5 = 400
The HCF is found by multiplying together all of the numbers in the overlapping quadrant. As there is only one number present:
HCF = 2