Prime factorisation: factor trees and the division method

Prime factorisation means writing a number as a product of prime numbers only. For example 60 = 2 × 2 × 3 × 5. Every whole number greater than 1 can be written this way, and apart from the order of the primes there is only one way to do it. This rule is called the fundamental theorem of arithmetic, and it is why prime factors are so useful: they are a number's fingerprint.

Method 1: the factor tree

Split the number into any two factors. Split each of those again. Stop a branch when it reaches a prime.

1. 60 splits into 6 × 10.
2. 6 splits into 2 × 3. Both are prime, so those branches end.
3. 10 splits into 2 × 5. Both are prime.
4. Collect the ends of the branches: 2, 3, 2 and 5.

So 60 = 2 × 2 × 3 × 5. If you had begun with 4 × 15 or 2 × 30, the tree would look different but the primes at the ends would be the same. Try it once: it is the best way to believe the rule.

Method 2: repeated division (the ladder)

Divide by the smallest prime that goes in exactly, and keep dividing the answer until you reach 1.

Read down the left side: 360 = 2 × 2 × 2 × 3 × 3 × 5. This method is tidier than the tree for large numbers, and it is the one most Indian textbooks use.

Writing the answer with powers

Repeated primes are gathered with a small raised number, called an index or power. 360 = 2³ × 3² × 5. Read it as "2 cubed times 3 squared times 5".

What prime factorisation is used for

• HCF: multiply the primes that two numbers share, each with its smaller power.
• LCM: multiply every prime that appears, each with its larger power.
• Counting factors: add 1 to each power and multiply. For 360 the powers are 3, 2 and 1, so it has 4 × 3 × 2 = 24 factors.
• Square roots: if every power is even, the number is a perfect square. 144 = 2⁴ × 3², so its square root is 2² × 3 = 12.
• Cube roots: if every power is a multiple of 3, the number is a perfect cube. 216 = 2³ × 3³, so its cube root is 2 × 3 = 6.
• Simplifying fractions: cancel the primes that top and bottom share.

Mistakes to avoid

• Stopping too early. 60 = 4 × 15 is a factorisation but not a prime one. Check that every number in the answer is prime.
• Using 1. 1 is not prime and is never written in a prime factorisation.
• Missing a repeated prime. After dividing by 2, try 2 again before moving to 3.
• Giving up on a large leftover. If nothing up to the square root of what is left divides it, the leftover is itself prime. Write it down and finish.

A harder example

For 1001: it is odd, its digits add to 2, and it does not end in 0 or 5, so 2, 3 and 5 fail. But 1001 ÷ 7 = 143. Then 143 ÷ 11 = 13, and 13 is prime. So 1001 = 7 × 11 × 13. That is why multiplying any three-digit number by 1001 writes it twice: 456 × 1001 = 456456.

The factor finder shows the ladder for any number you type.

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