Highest Common Factor definition
The highest common factor, often shortened to HCF, is the largest factor shared by two or more numbers. A factor is a number that divides exactly into another number with no remainder. A common factor is a factor that two or more numbers have in common. The highest common factor is the biggest of these shared factors.
For example, the factors of 18 are 1, 2, 3, 6, 9 and 18. The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. The common factors are 1, 2, 3 and 6. The highest common factor is therefore 6. This means 6 is the largest number that divides exactly into both 18 and 24.
Highest common factor is used in number, fractions, ratio, proportion, divisibility, prime factorisation, algebra and problem solving. At GCSE level, students use HCF to simplify fractions, split quantities into the largest equal groups, solve sharing problems and compare number structures. It is often studied together with lowest common multiple, because both ideas use factors and multiples but in different ways.
One method for finding the HCF is to list all the factors of each number. This works well for smaller numbers. For example, to find the HCF of 20 and 30, list the factors of 20 as 1, 2, 4, 5, 10 and 20. List the factors of 30 as 1, 2, 3, 5, 6, 10, 15 and 30. The common factors are 1, 2, 5 and 10, so the HCF is 10.
Another method uses prime factorisation. For example, 60 can be written as 2 times 2 times 3 times 5. The number 90 can be written as 2 times 3 times 3 times 5. The shared prime factors are 2, 3 and 5, so the HCF is 2 times 3 times 5, which equals 30. Prime factorisation is especially useful for larger numbers because it gives a systematic method.
HCF is very useful for simplifying fractions. The fraction 18 over 24 can be simplified by dividing both the numerator and denominator by their HCF, which is 6. This gives 3 over 4. Because 6 is the highest common factor, the fraction is simplified fully in one step. If a smaller common factor is used first, the fraction may need more than one simplification step.
HCF can also appear in worded problems. If you have 36 red beads and 48 blue beads and want to make identical packs with no beads left over, the largest number of identical packs is the HCF of 36 and 48. The HCF is 12, so you can make 12 packs. Each pack would contain 3 red beads and 4 blue beads. This shows how HCF helps with equal grouping and sharing.
Related ideas include factors, factor pairs, common factors, prime factors, multiples, lowest common multiple, simplifying fractions and ratio. A clear definition of highest common factor helps students understand that HCF is about the largest shared divisor. In exams, always check that the factor divides into every number in the question and that there is no larger shared factor.