Prime Factors, HCF and LCM

Grade 4

Prime factor decomposition

Prime factor decomposition means writing a number as a product of its prime factors. Every integer greater than 1 can be written this way, and the result is unique — which is why it's such a useful tool.

The most reliable method is a factor tree: split the number into any two factors, then keep splitting each branch until every number at the end is prime. Circle the primes as you reach them.

It doesn't matter which pair of factors you start with — you'll always reach the same prime factors at the end. Start with an obvious one (e.g. split an even number by 2) to keep things tidy.

Worked example — factor tree

EXAMPLE

Write 60 as a product of its prime factors.

Split 60 → 6 × 10. Split 6 → 2 × 3 (both prime). Split 10 → 2 × 5 (both prime). Collect the primes: 2, 3, 2, 5. Write in order: 60 = 2² × 3 × 5. So 60 = 2² × 3 × 5.

Using prime factors to find HCF and LCM

Once you have both numbers in prime factor form, finding the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) becomes straightforward.

HCF: take the prime factors that appear in both numbers, using the lower power for each. Multiply them together.

LCM: take all prime factors that appear in either number, using the higher power for each. Multiply them together.

EXAMPLE

Find the HCF and LCM of 60 and 84.

60 = 2² × 3 × 5 and 84 = 2² × 3 × 7. HCF — factors in both: 2² and 3 (5 and 7 don't appear in both). HCF = 2² × 3 = 4 × 3 = 12. LCM — all factors, higher power: 2², 3, 5, 7. LCM = 4 × 3 × 5 × 7 = 420.

Watch out

Don't confuse HCF and LCM. The HCF is always smaller than or equal to both numbers; the LCM is always larger than or equal to both. If your HCF comes out bigger than either number, something has gone wrong.

Try this one

Without using a calculator, simplify fully \( \dfrac{4^{3} \times 8^{2}}{2^{10}} \).

Give your answer as a power of 2. [3]

Nova's hint:

This uses index laws with the same base.

Ask yourself: 'can I write every number in the expression as a power of 2?' Once all bases match, use the multiplication and division index laws: \( a^m \times a^n = a^{m+n} \) and \( a^m \div a^n = a^{m-n} \).

Try to Solve it with Nova?