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*.

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.

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

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