Prime factor decomposition

Prime factor decomposition means breaking a number down into the prime numbers that multiply to make it. For example, 252 = 2 × 2 × 3 × 3 × 7, which is written in index form as 2² × 3² × 7. It is also called writing a number as a product of prime factors, or prime factorisation (prime factorization in the US). Every whole number greater than 1 has exactly one prime factor decomposition.

What is prime factor decomposition?

A prime number has exactly two factors, 1 and itself. The first few are 2, 3, 5, 7, 11 and 13 (1 is not prime). Every other whole number above 1 can be built by multiplying primes together, and there is only one way to do it, apart from the order. This is the unique factorisation theorem, and it is why prime factors are so useful: they are like a number's fingerprint.

How to find the prime factor decomposition

Method 1: a factor tree

Split the number into any two factors. Keep splitting each branch until it ends in a prime, and circle the primes. For 252:

Collect the circled primes and write them in order, smallest first: 252 = 2 × 2 × 3 × 3 × 7 = 2² × 3² × 7. You could start with 2 × 126 or 3 × 84 instead; any tree gives the same primes.

Method 2: repeated division

Divide by the smallest prime that goes in exactly, and keep going until you reach 1. For 540:

The primes you divided by give 540 = 2 × 2 × 3 × 3 × 3 × 5 = 2² × 3³ × 5.

Writing the answer in index form

Questions often say "give your answer in index form". Group the repeated primes as powers: 2 × 2 × 2 × 5 × 5 = 2³ × 5². Always check by multiplying back: 8 × 25 = 200.

To see the factor tree for any number, use the Prime Factorisation Calculator. The prime factorization calculator guide shows how to use it, and how to count a number's factors from its prime factors.

Check your understanding.

HCF and LCM using prime factors

Prime factors make the highest common factor (HCF) and lowest common multiple (LCM) quick, especially for large numbers. Take 252 = 2² × 3² × 7 and 120 = 2³ × 3 × 5.

Using a Venn diagram

Put the primes the two numbers share in the overlap, and the rest in each circle. 252 has 2, 2, 3, 3, 7 and 120 has 2, 2, 2, 3, 5. They share 2, 2 and 3:

• HCF = multiply the primes in the overlap: 2 × 2 × 3 = 12.
• LCM = multiply every prime in the diagram: 3 × 7 × 2 × 2 × 3 × 2 × 5 = 2520.

Using index form

For the HCF, take the lowest power of each prime that appears in both numbers: 2² and 3¹, so HCF = 2² × 3 = 12. For the LCM, take the highest power of every prime that appears in either: 2³, 3², 5 and 7, so LCM = 2³ × 3² × 5 × 7 = 2520.

Check: HCF × LCM = 12 × 2520 = 30 240, and 252 × 120 = 30 240. For two numbers this always works.

Using prime factors with powers and roots

Square and cube numbers

A number is a square number when every power in its prime factor decomposition is even, and a cube number when every power is a multiple of 3.

• 1764 = 2² × 3² × 7². Every power is even, so it is square: √1764 = 2 × 3 × 7 = 42.
• 3375 = 3³ × 5³. Every power is a multiple of 3, so it is a cube: ∛3375 = 3 × 5 = 15.

Example: making a square number

What is the smallest whole number you can multiply 252 by to make a square number?

252 = 2² × 3² × 7. The 7 has an odd power, so multiply by one more 7: 252 × 7 = 1764 = 2² × 3² × 7², which is 42². The answer is 7.

Example: working with index form

Given that A = 2³ × 5, write A² as a product of prime factors. Squaring doubles every power: A² = 2⁶ × 5². Check: A = 40 and 40² = 1600 = 64 × 25.

Common mistakes

• Stopping too early. Every branch must end in a prime: 252 = 4 × 63 isn't finished.
• Including 1. 1 is not prime, so it never appears in a prime factor decomposition.
• Losing a factor. Multiply your answer back to check it gives the original number.
• Writing 2 × 3 = 2³. A power counts how many times the same prime appears.
• Swapping HCF and LCM. The HCF uses the overlap (lowest powers); the LCM uses everything (highest powers).
• Listing a non-prime. 9 and 15 are not prime: split them further.

Practice questions

1. Write 180 as a product of prime factors in index form.
2. Find the HCF and LCM of 180 and 252.
3. Use prime factors to find √4356, given that 4356 = 2² × 3² × 11².
4. What is the smallest number you can multiply 98 by to make a square number?

Answers: 1) 180 = 2² × 3² × 5. 2) HCF = 2² × 3² = 36; LCM = 2² × 3² × 5 × 7 = 1260. 3) 2 × 3 × 11 = 66. 4) 98 = 2 × 7², so multiply by 2: 196 = 14².

Practise with a short timed paper

Primes, factors and multiples come up on both tiers of GCSE Maths. The daily GCSE Maths Higher and GCSE Maths Foundation papers are short timed papers of 8 questions, with a new one every day.

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Maths · NumberPrime factorization calculator with a factor treeFree prime factorization calculator: break any whole number up to a trillion into primes, with a factor tree, index form and the number of factors.