Integers
Highest common factors
The mirror image of the LCM. List what divides into each number, find what they share, and take the biggest — then use it to simplify a fraction in one step.
Factors
The factors of a number divide into it exactly. The factors of 18 are 1, 2, 3, 6, 9 and 18. The factors of 27 are 1, 3, 9 and 27. A good way to find them is in pairs: 18 = 1 × 18 = 2 × 9 = 3 × 6.
Common factors and the HCF
The numbers appearing in BOTH lists are common factors. The common factors of 18 and 27 are 1, 3 and 9. The biggest of them is the HIGHEST COMMON FACTOR (HCF). The HCF of 18 and 27 is 9.
A worked example
Find the HCF of 28 and 42. 28 = 1 × 28 = 2 × 14 = 4 × 7, so the factors are 1, 2, 4, 7, 14, 28. 42 = 1 × 42 = 2 × 21 = 3 × 14 = 6 × 7, so the factors are 1, 2, 3, 6, 7, 14, 21, 42. Common factors: 1, 2, 7 and 14. The HCF is 14.
Using the HCF to simplify fractions
This is where the HCF earns its keep. The HCF of 16 and 40 is 8. To write 16/40 as simply as possible, divide BOTH numbers by 8: 16 ÷ 8 = 2 and 40 ÷ 8 = 5, so 16/40 = 2/5. Dividing by the HCF simplifies the fraction as far as it will go, in one step.
Worked examples
The highest common factor of 18 and 27
- Factors of 18: 1, 2, 3, 6, 9, 18.
- Factors of 27: 1, 3, 9, 27.
- Common: 1, 3, 9. The biggest is 9.
Answer: 9
The HCF of 28 and 42
- Factors of 28: 1, 2, 4, 7, 14, 28.
- Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42.
- Common: 1, 2, 7, 14.
Answer: 14
Simplifying 16/40 with the HCF
- The HCF of 16 and 40 is 8.
- 16 ÷ 8 = 2 and 40 ÷ 8 = 5.
Answer: 2/5, in one step
Finding factors in pairs
- 18 = 1 × 18, 2 × 9, 3 × 6.
- Each pair gives you two factors at once.
Answer: 1, 2, 3, 6, 9, 18
The HCF of 12 and 18
- Factors of 12: 1, 2, 3, 4, 6, 12.
- Factors of 18: 1, 2, 3, 6, 9, 18.
- The biggest shared one is 6.
Answer: 6
The HCF of 7 and 9
- 7 is prime, so its factors are 1 and 7.
- 9 has factors 1, 3, 9.
- The only shared factor is 1.
Answer: 1 — the numbers share nothing else
Where you see this in real life
- You have 24 pencils and 36 rubbers and want identical packs with none left over. The HCF, 12, is the largest number of packs you can make.
- Simplifying 16/40 down to 2/5 in one step only works if you divide by the HCF — divide by a smaller common factor and you have to go again.
Key words
- Factor
- a number that divides into another exactly
- Common factor
- a number that is a factor of two or more numbers
- Highest common factor (HCF)
- the largest common factor of two numbers
- Prime number
- a number with exactly two factors: 1 and itself
- Simplify
- write a fraction with the smallest possible numbers