Appendix A

A:02G HCF and LCM by Prime Factors
When we express numbers as products of their prime factors it is sometimes easier to find their
HCF and LCM.
For the numbers 400 and 1080 find:
1 the HCF, and 2 the LCM.
400
20 20
4 5 5 4
2 2 5 5 2 2
worked examples
1080
20 54
4 5 6 9
2 2 5 2 3 3 3
From these factor trees we find:
Number
The two numbers have 2 × 2 × 2 × 5 in common.
2 × 2 × 2 × 5 = 2 3 × 5 or 40.
∴ HCF = 40
The highest common factor of 400 and 1080 is 40.
The LCM must contain the whole of both numbers, so it will contain the whole of 400
(ie 2 × 2 × 2 × 2 × 5 × 5) as well as any other prime factors of 1080 not already written down.
LCM = (2 × 2 × 2 × 2 × 5 × 5) × (3 × 3 × 3)
= 2 4 × 3 3 × 5 2
= 10 800
The lowest common multiple of 400 and 1080 is 10 800.
■ Note:
The HCF is found by using the smallest common power of each prime factor of the numbers.
The LCM is found by using the highest power of each prime factor of the numbers.
400
1080
Product of prime factors
2 × 2 × 2 × 2 × 5 × 5
or 2 4 × 5 2
2 × 2 × 2 × 3 × 3 × 3 × 5
or 2 3 × 3 3 × 5
A:02H Divisibility Tests
If 4 divides a number and leaves no remainder, then the number is divisible by 4. Here are some
simple divisibility tests you should know.
Divisor Divisibility test Example
2 The number is even,
ie it ends in 0, 2, 4, 6 or 8.
3 The sum of the digits is
divisible by 3.
4 The number formed by the last
two digits is divisible by 4.
4136 is divisible by 2
as it is even.
30 012 is divisible by 3
as (3 + 0 + 0 + 1 + 2) is
divisible by 3.
76 112 is divisible by 4
as 12 is divisible by 4.
5 The last digit is 5 or 0. 11 225 is divisible by 5
as it ends with a 5.
Learn these!
They’re useful.
8 NEW SIGNPOST MATHEMATICS 8