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Pioneers Maths

Maths · Number & Negatives

Factors, multiples & primes

HCF, LCM, and prime factorisation.

Not startedCAMBRIDGE · 7CBSE · VIIB · MYP-1

Factors and Multiples

Factors divide exactly into a number with no remainder. Multiples are what you get when you multiply a number. For example, factors of 12 are 1, 2, 3, 4, 6, 12. Multiples of 5 are 5, 10, 15, 20, and so on.

Prime Numbers and Prime Factorisation

A prime number has exactly two factors: 1 and itself. Examples include 2, 3, 5, 7, 11, 13. Prime factorisation means writing a number as a product of prime numbers only. For instance, 12 = 2 × 2 × 3.

Highest Common Factor (HCF)

The HCF is the largest number that divides exactly into two or more numbers. To find it, list all factors of each number, then pick the biggest one they share. The HCF of 12 and 18 is 6.

Lowest Common Multiple (LCM)

The LCM is the smallest number that is a multiple of two or more numbers. List multiples of each number until you find the smallest one they share. The LCM of 4 and 6 is 12.

Using Prime Factorisation for HCF and LCM

Write each number as a product of primes. For HCF, multiply the common prime factors. For LCM, multiply the highest power of each prime that appears. This method works especially well for larger numbers.

Worked examples

  1. 1. Find the HCF of 24 and 36 using prime factorisation.

    1. Write 24 as prime factors: 24 = 2 × 2 × 2 × 3
    2. Write 36 as prime factors: 36 = 2 × 2 × 3 × 3
    3. Identify common prime factors: 2 × 2 × 3
    4. Multiply them together: 2 × 2 × 3 = 12
    5. The HCF of 24 and 36 is 12.

  2. 2. Find the LCM of 6 and 8.

    1. Write 6 as prime factors: 6 = 2 × 3
    2. Write 8 as prime factors: 8 = 2 × 2 × 2
    3. For LCM, take the highest power of each prime: 2³ and 3¹
    4. Multiply them: 2 × 2 × 2 × 3 = 24
    5. The LCM of 6 and 8 is 24.

  3. 3. Write 60 as a product of prime factors.

    1. Start by dividing by the smallest prime, 2: 60 ÷ 2 = 30
    2. Continue dividing by 2: 30 ÷ 2 = 15
    3. 15 is not divisible by 2, try 3: 15 ÷ 3 = 5
    4. 5 is prime, so we stop
    5. 60 = 2 × 2 × 3 × 5

Try it yourself

Find the HCF of 18 and 30.

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