Practice Maths

L40 — Prime Factors

Key Terms

prime number
A whole number greater than 1 with exactly two factors: 1 and itself. Primes up to 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
composite number
A whole number greater than 1 that has more than two factors. E.g. 12 is composite because it has factors 1, 2, 3, 4, 6, 12.
prime factorisation
Writing a number as a product of its prime factors. E.g. 36 = 22 × 32. Every composite number has exactly one prime factorisation.
factor tree
A diagram used to find the prime factorisation of a number by repeatedly splitting it into factor pairs until all branches end at primes.
HCF (highest common factor)
The largest factor shared by two or more numbers. Found using prime factorisation by taking each shared prime to the lower power.
LCM (lowest common multiple)
The smallest number that is a multiple of two or more numbers. Found using prime factorisation by taking each prime to the highest power.
perfect square
A number whose square root is a whole number. In its prime factorisation, all exponents are even. E.g. 36 = 22 × 32 is a perfect square because √36 = 6.

Prime Factorisation Using a Factor Tree

  1. Write the number at the top.
  2. Split into any two factors (not 1 × the number). Branch them below.
  3. Keep factoring each branch until every end is a prime number.
  4. Collect all the primes at the ends of the branches.
  5. Write using index notation where primes repeat.
Hot Tip: 1 is not a prime number — it has only one factor, but primes must have exactly two. Every composite number also has a unique prime factorisation (the Fundamental Theorem of Arithmetic).

Worked Examples

Prime factorisation of 48:

48 = 2 × 24 = 2 × 2 × 12 = 2 × 2 × 2 × 6 = 2 × 2 × 2 × 2 × 3

Answer: 48 = 24 × 3

Prime factorisation of 90:

90 = 2 × 45 = 2 × 3 × 15 = 2 × 3 × 3 × 5

Answer: 90 = 2 × 32 × 5

HCF(12, 18): 12 = 22 × 3, 18 = 2 × 32. Common: 21 × 31 = 6

LCM(12, 18): Take highest powers: 22 × 32 = 36

Primes: The Atoms of Arithmetic

Prime numbers are the building blocks of all whole numbers — just like atoms are the building blocks of all matter. Every number greater than 1 is either prime itself, or can be built by multiplying primes together. This is called the Fundamental Theorem of Arithmetic: every number has exactly one prime factorisation.

The primes under 30 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Notice that 2 is the only even prime — every other even number is divisible by 2, so not prime. And 1 is NOT prime, because it has only one factor (itself), while primes need exactly two.

Building the Factor Tree

It doesn’t matter which factors you start with — you’ll always end with the same prime factorisation. For example:

  • 36 = 4 × 9 = 2 × 2 × 3 × 3 = 22 × 32
  • 36 = 6 × 6 = 2 × 3 × 2 × 3 = 22 × 32

Both routes give the same answer. The factorisation is unique.

Remember: A factor tree is only complete when every leaf is prime. Stopping at 36 = 4 × 9 is wrong — 4 and 9 are not prime! Keep going: 4 = 2 × 2 and 9 = 3 × 3.

Using Prime Factorisation: HCF and LCM

Prime factorisation is the most reliable method for finding the HCF and LCM of any two numbers:

  • HCF: take the prime factors that appear in both numbers, with the lower power. HCF(12, 18): 12 = 22 × 3, 18 = 2 × 32. Both share 21 and 31. HCF = 6.
  • LCM: take all prime factors, with the higher power. LCM(12, 18): highest powers are 22 and 32. LCM = 36.

Why Prime Numbers Matter in the Real World

Every time you log in to a website securely (https), your data is protected using a system called RSA encryption. RSA relies on the fact that multiplying two large prime numbers together is easy, but factoring the result back into those primes takes so long that even computers can’t do it in a reasonable time. The security of online banking depends on prime factorisation!

Common Mistake: When finding HCF, students sometimes take the highest power of each prime instead of the lowest. For HCF, always take the lower (or equal) power — it’s the opposite of LCM.

  1. Identify Prime Numbers

    State whether each number is prime or composite. If composite, give one factor other than 1 and itself.

    1. 7
    2. 9
    3. 11
    4. 15
    5. 17
    6. 21
    7. 23
    8. 27

  2. Complete Factor Trees

    Draw a factor tree and write the prime factorisation for each number.

    1. 12
    2. 20
    3. 36
    4. 48
    5. 60
    6. 72
    7. 84
    8. 100

  3. Write Prime Factorisation in Index Form

    Write the prime factorisation of each number using index notation.

    1. 18
    2. 24
    3. 32
    4. 40
    5. 54
    6. 75
    7. 90
    8. 120

  4. Highest Common Factor (HCF)

    Use prime factorisation to find the HCF of each pair.

    1. HCF(12, 18)
    2. HCF(24, 36)
    3. HCF(30, 45)
    4. HCF(48, 72)
    5. Explain how you use prime factorisations to find the HCF.

  5. Lowest Common Multiple (LCM)

    Use prime factorisation to find the LCM of each pair.

    1. LCM(4, 6)
    2. LCM(8, 12)
    3. LCM(5, 7)
    4. LCM(6, 10)
    5. Explain how you use prime factorisations to find the LCM.

  6. Problem Solving with Prime Factors

    1. A number has prime factorisation 23 × 3 × 7. What is the number?
    2. A number has prime factorisation 22 × 52. What is the number?
    3. Two bells ring at intervals of 12 minutes and 18 minutes. Both ring at 9:00 am. When will they next ring together? (Hint: use LCM.)
    4. Explain why 1 is not considered a prime number. What would go wrong with prime factorisation if 1 were prime?
    5. Is 24 × 32 a perfect square? Explain using the prime factorisation.

  7. Larger Numbers — Factor Trees and Factorisations

    Find the prime factorisation of each number. Write your answer in index form.

    1. 126
    2. 144
    3. 180
    4. 252
    5. 360
    6. 500
    7. 630
    8. 1000

  8. HCF and LCM of Three Numbers

    Use prime factorisation to find the HCF and LCM of each set of three numbers.

    1. HCF(12, 18, 24)
    2. LCM(4, 6, 9)
    3. HCF(30, 45, 60)
    4. LCM(6, 8, 12)
    5. A number has both 12 and 18 as factors. What is the smallest possible value of the number?

  9. Perfect Squares via Prime Factorisation

    Recall: A perfect square has all even exponents in its prime factorisation. For example, 22 × 32 = 36 is a perfect square because √36 = 6. If any exponent is odd, it is not a perfect square.
    1. A number has prime factorisation 22 × 32 × 52. Explain why it is a perfect square, then work out its value.
    2. 72 = 23 × 32. What is the smallest whole number you can multiply 72 by to get a perfect square? Show your working.
    3. A number has prime factorisation 26 × 34. Is it a perfect square? Work out its value.
    4. Which of these are perfect squares? Explain how you can tell from the exponents alone.
        (i) 24 × 32     (ii) 23 × 52     (iii) 22 × 74
    5. Find the HCF of 24 × 33 × 5 and 23 × 3 × 52 × 7. (Hint: take the lower power of each prime that appears in both numbers.)

  10. Real-World Applications of HCF and LCM

    1. A baker makes batches of 18 muffins. Another baker makes batches of 24. What is the fewest number of muffins they must each make so that both have an equal number?
    2. Tiles measuring 12 cm by 18 cm are to be cut into equal-sized smaller squares with no waste. What is the largest possible size for the small squares?
    3. Three buses leave a depot together. Bus A departs every 20 minutes, Bus B every 30 minutes, and Bus C every 45 minutes. How long until all three buses depart together again?
    4. A rectangular garden is 48 m long and 36 m wide. A gardener wants to plant trees at equal intervals along the perimeter, with trees at every corner. What is the greatest distance between consecutive trees?
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