Prime Factors, HCF & LCMCambridge IGCSE Maths: Revision notes

A prime number has exactly two factors: 1 and itself (e.g. 2, 3, 5, 7, 11, 13...). A prime factor of a number is a prime number that divides into it exactly.

Every integer greater than 1 can be written as a unique product of prime factors — this is called prime factor decomposition.

Use a factor tree or repeated division by the smallest prime.

Example: express 72 as a product of prime factors

1. 72 ÷ 2 = 36
2. 36 ÷ 2 = 18
3. 18 ÷ 2 = 9
4. 9 ÷ 3 = 3
5. 3 ÷ 3 = 1

So 72 = $2^3 \times 3^2$

The HCF of two or more numbers is the largest number that divides exactly into all of them.

Using prime factor decomposition:

1. Write each number as a product of prime factors
2. Identify the prime factors common to all numbers
3. Multiply together the LOWEST power of each common prime factor

Example: HCF of 36 = $2^2 \times 3^2$ and 60 = $2^2 \times 3 \times 5$. Common factors: $2^2$ and $3^1$. HCF = $4 \times 3 = 12$

The LCM of two or more numbers is the smallest number that is a multiple of all of them.

Using prime factor decomposition:

1. Write each number as a product of prime factors
2. Take every prime factor that appears in any number
3. Multiply together the HIGHEST power of each prime factor

Example: LCM of 36 = $2^2 \times 3^2$ and 60 = $2^2 \times 3 \times 5$. Highest powers: $2^2$, $3^2$, $5^1$. LCM = $4 \times 9 \times 5 = 180$