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How to Learn Prime Factorisation Quickly and Correctly | Quick Notes

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eduKate Asia
Three students sit around open books and worksheets at a classroom table, reading, writing and discussing the work together.

To learn prime factorisation quickly and correctly, remember that every whole number greater than 1 can be broken into a product of prime numbers in a way that is unique apart from the order of the factors. A prime number has exactly two positive factors: 1 and itself. Once you understand primes, a simple factor tree or repeated division shows exactly which prime building blocks make up a number. These eduKateSG Mathematics Quick Notes teach an easy method, show how to write index form and explain how to check every answer.

  • Recognise primes: a prime has exactly two positive factors.
  • Start with easy divisors: test 2, 3 and 5 before trying larger primes.
  • Split a composite number: write it as a product of two smaller factors.
  • Continue until all leaves are prime: do not stop at a composite factor.
  • Group repeats: express repeated primes using indices.
  • Check: multiply the prime factors to recover the original number.

First, make sure you know which numbers are prime. The first few are 2, 3, 5, 7, 11, 13, 17 and 19. The number 2 is the only even prime number. The number 1 is not prime, because it has just one positive factor, not exactly two. A composite number is a positive integer greater than 1 with more than two positive factors; for example, 12 is composite because 1, 2, 3, 4, 6 and 12 all divide it exactly. Distinguish a prime from a composite before building a factor tree.

Second, try an ordinary factor tree. For 60, split 60 into 6 × 10. Then split 6 into 2 × 3 and 10 into 2 × 5. The leaves are 2, 3, 2 and 5, all primes. Therefore 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5. It is fine to start with a different pair, such as 3 × 20, because if you finish correctly, the same prime factors appear. Rewriting repeated factors in index form makes the final answer neat.

Third, check factorisation by multiplying back. If your answer for 60 is 2² × 3 × 5, calculate 4 × 3 × 5 = 60. If the product does not equal the original number, a prime factor was lost or repeated. If a branch ends with 4, 6, 9, 10 or another composite number, continue splitting it. The aim is not just to find any multiplication sentence, but to express the whole number entirely as a product of primes.

Worked example: factorise 84

Begin with 84 ÷ 2 = 42. Then 42 ÷ 2 = 21. Since 21 = 3 × 7, all the final factors are prime. Write 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7. Check: 4 × 3 × 7 = 84. A useful format is a vertical list of successive divisions: 84 → 42 → 21 → 7 → 1, with the divisors 2, 2, 3 and 7 beside it.

Worked example: factorise 360 in index form

Divide by 2 repeatedly: 360 ÷ 2 = 180, 180 ÷ 2 = 90, and 90 ÷ 2 = 45. Then 45 ÷ 3 = 15, 15 ÷ 3 = 5 and 5 ÷ 5 = 1. The prime divisors are 2, 2, 2, 3, 3, 5. Therefore 360 = 2³ × 3² × 5. Check: 8 × 9 × 5 = 360. Each index records how many times the corresponding prime appears.

When can you stop testing a possible prime?

To check whether a positive integer n greater than 1 is prime, it is enough to test prime divisors up to √n. Why? If n had a pair of factors larger than √n, their product would exceed n; therefore any composite n has a factor no bigger than √n. For example, √29 is a little more than 5, so test prime divisors 2, 3 and 5. None divides 29, so 29 is prime. You do not have to test every number below 29.

Factor tree or repeated division: which is easier?

  • Factor tree: visual and friendly; start with any correct factor pair and split until all branches are prime.
  • Repeated division: systematic; divide by the smallest useful prime and continue with the quotient.
  • Index form: compact final presentation, such as 72 = 2³ × 3².
  • Multiplication check: essential whichever method you use.

Why prime factorisation helps with other topics

Prime factors make highest common factors and lowest common multiples easier to calculate. They also help simplify fractions. For example, 18/24 can be simplified because 18 = 2 × 3² and 24 = 2³ × 3. Both contain the factor 2 × 3 = 6, so dividing numerator and denominator by 6 gives 3/4. Prime factorisation helps you see the shared building blocks directly rather than guessing a simplification.

Important special cases

The number 1 has no prime factorisation as a product of primes; it is the empty-product case and is not itself prime. The number 0 is not represented by the usual unique prime factorisation of positive integers greater than 1. Negative integers can be written as −1 times prime factors of their positive magnitude, but school prime factorisation questions typically ask about positive whole numbers. Start with the positive integer the question actually names.

Common mistakes and how to repair them

  • Calling 1 prime: a prime has exactly two positive factors.
  • Stopping at 6 or 9: those are composite; split them further.
  • Writing 2³ as 2 × 3: an index counts repeated multiplication, so 2³ = 8.
  • Missing a prime after dividing: record every divisor at each step.
  • Changing the answer when the factor tree starts differently: completed prime factors are unique up to order.
  • Not multiplying back: always verify the product equals the starting number.

Mini-test: prime factorise these numbers

  1. Write 72 as a product of prime factors in index form.
  2. Write 150 as a product of prime factors in index form.
  3. Write 98 as a product of prime factors in index form.
  4. Explain whether the number 1 is prime, composite or neither.

Answers with a multiplication check

1. 72 = 2³ × 3², since 8 × 9 = 72. 2. 150 = 2 × 3 × 5², since 2 × 3 × 25 = 150. 3. 98 = 2 × 7², since 2 × 49 = 98. 4. 1 is neither prime nor composite, because it has only one positive factor.

A ten-minute study routine

  1. Minutes 1–2: list prime numbers below 20 and explain why 1 is not prime.
  2. Minutes 3–5: build one factor tree and circle every prime leaf.
  3. Minutes 6–7: rewrite the factors in index form.
  4. Minute 8: multiply the factors to check the original number.
  5. Minutes 9–10: factorise a new number without looking at the example.

The reliable shortcut is a complete method, not a guess: split, continue until prime, group equal factors and multiply back. Once that routine feels natural, HCF, LCM and simplifying fractions become more understandable.

Learn the next connected idea

Use these eduKateSG Mathematics Quick Notes to build on today’s learning: Factors and Multiples · HCF and LCM · Indices · Fractions. Complete a fresh question from memory before moving on.

The longer teaching reference, Secondary 1 Mathematics Tutor Clementi | Small Groups Tutorials, remains unchanged. This is a separate Quick Notes lesson.