Highest Common Factor

Highest Common Factor

GCSE(F), GCSE(H),

The Highest Common Factor, or HCF, is the largest factor that is shared by two integers. For smaller numbers, the HCF can be determined by writing out a list of factors for each number, and determining the highest number which is on both lists:

The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30

The factors of 45 are 1, 3, 5, 9, 15 and 45.

The highest number that is on both lists is 15: this is the Highest Common Factor (or HCF).

For larger numbers, work out the Products of the Primes for both numbers using a prime factor tree. Draw a Venn Diagram, with each set containing the factors for each number. The shared factors (intersection), when multiplied together, will give the HCF. Higher students: The intersection could be written as factors of 30 ∩ factors of 45.

Examples

1. What is the highest common factor of 60 and 84?

Answer: 12

60: 2 x 2 x 3 x 5

84: 2 x 2 x 3 x 7

Factors in both sets are 2, 2 and 3: 2 x 2 x 3 = 12.

Venn diagram showing HCF of 60 and 84

2. What is the highest common factor of 63 and 120?

Answer: 3

63: 3 x 3 x 7

120: 2 x 2 x 2 x 3 x 5

There is only one factor in both sets: 3