A common factor divides both numbers exactly

1 is a factor of every whole number, so any two numbers have at least one common factor. The largest of them is the highest common factor, written HCF.

The HCF is also called the greatest common divisor, GCD: a divisor of a number is a factor of it, so the two names mean the same thing.

While the factor lists are short, listing every factor of each number finds the HCF:

The HCF is everything the two numbers share

Match equal prime factors in 360 and 756. Each matched pair belongs in a factor of both numbers, so multiplying all the matched primes gives the greatest common factor.

Index form lets the primes be compared one at a time

360 and 756 each have 24 factors. The two lists run to 48 numbers before the common ones can be compared. Written in index form, each index counts how many of one prime factor the number holds, and two indices are compared instead of two lists.

1. Factorise both numbers into index form, with prime bases in ascending order.
2. Compare the two indices of each prime, counting a missing prime as index 0, and take the lower.
3. Multiply the chosen prime powers. A prime whose lower index is 0 contributes no factor.

Take each shared prime to its lower index

A factor of 360 is built from the primes of 360. It contains no more factors of each prime than 360 does: 2^² × 5 = 20 is a factor of 360, but 2^⁴ is not, because 360 contains only three factors of 2.

A factor of both 360 and 756 holds no more of each prime than either does, so the largest takes each prime at the lower index: two 2s and two 3s, 2^² × 3^² = 36.

A power with an index of 0 is 1, which is why writing 7^⁰ leaves 360 unchanged.

Every common factor is a factor of the HCF

The common factors of 360 and 756 are exactly the factors of 36: each is built from 2s and 3s with indices no higher than 2, so each divides 2^² × 3^².

Find the HCF of two chosen numbers

Common mistakes

• Taking the higher index For 90 = 2 × 3^² × 5 and 84 = 2^² × 3 × 7, taking the higher index of each shared prime gives 2^² × 3^² = 36, which divides neither: 90 = 2 × 36 + 18 and 84 = 2 × 36 + 12. The higher index belongs to the lowest common multiple, not the HCF.
• Keeping a prime that only one number has For 90 and 84, including the 5 and the 7 gives 2 × 3 × 5 × 7 = 210. That is larger than both 90 and 84, so it cannot be a factor of either; the HCF is 2 × 3 = 6.
• Stopping at the first common factor found 5 divides both 20 and 50, but so does 10. The HCF is 10, twice the 5 that was written down.

Summary

1. A common factor divides both numbers exactly; the HCF is the largest common factor.
2. While the factor lists are short, list the factors of each number and pick the largest that appears in both lists.
3. For larger numbers, write both in index form; the HCF is every shared prime, each to its lower index: “find the highest common factor of 360 and 756” is answered 36.
4. Leave out any prime that appears in only one of the numbers; with no shared prime, the HCF is 1.
5. Every common factor is a factor of the HCF: the common factors of 360 and 756 are the factors of 36.
6. Divide both numbers by the HCF: the quotients share no prime.