Factors, Multiples & Primes
Finding HCF and LCM, identifying prime numbers, square numbers and cube numbers.
What is Factors, Multiples & Primes?
Factors are numbers that divide exactly into another number. Multiples are the results of multiplying a number by whole numbers. Prime numbers have exactly two factors: 1 and themselves.
In the 11+ exam, you may be asked to list factors, find common factors or multiples, identify prime numbers, or work with square and cube numbers. These skills also help with simplifying fractions and solving problems.
Step-by-Step Method
To find factors, work in pairs
Start from 1 and work up. 1 x 36 = 36, 2 x 18 = 36, 3 x 12 = 36, 4 x 9 = 36, 6 x 6 = 36. Stop when the pairs meet.
To find the HCF, list factors of both numbers
The highest common factor is the largest number that appears in both lists.
To find the LCM, list multiples of both numbers
The lowest common multiple is the smallest number that appears in both lists.
To check if a number is prime, test divisibility
Try dividing by 2, 3, 5, 7 and so on. You only need to test up to the square root of the number.
Know your square and cube numbers
Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Cube numbers: 1, 8, 27, 64, 125.
Worked Examples
Find all the factors of 36.
Working
- 1 x 36 = 36
- 2 x 18 = 36
- 3 x 12 = 36
- 4 x 9 = 36
- 6 x 6 = 36
- Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36.
Find the LCM of 6 and 8.
Working
- Multiples of 6: 6, 12, 18, 24, 30…
- Multiples of 8: 8, 16, 24, 32…
- The first number in both lists is 24.
Is 51 a prime number?
Working
- Check: 51 / 2 = 25.5 (not a factor).
- 51 / 3 = 17 (this divides exactly!).
- Since 51 = 3 x 17, it has more than two factors.
Common Mistakes
Forgetting to include 1 and the number itself as factors.
1 and the number are always factors. 1 x n = n for every number.
Thinking 1 is a prime number.
1 is NOT prime because it only has one factor (itself). Primes must have exactly two factors.
Missing factor pairs by not working systematically.
Always start from 1 and work up in order, writing pairs. Stop when the pairs meet.
Top Tips
- Primes up to 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. The only even prime is 2.
- Divisibility shortcuts: even numbers divide by 2, numbers whose digits sum to a multiple of 3 divide by 3, numbers ending in 0 or 5 divide by 5.
- The HCF is useful for simplifying fractions. The LCM is useful for finding common denominators.
- Square numbers always have an odd number of factors (because the square root only counts once).
Ready to practise?
Put these techniques into action with our free practice papers.
Practise Maths Questions