Factors, Multiples and Primes –

Factors, multiples, and primes form the foundation of number work at GCSE. These concepts appear directly on exam papers and also underpin topics such as HCF, LCM, fractions, and algebra.

What Are Factors, Multiples and Primes?

A factor of a number divides into it exactly with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. Factors come in pairs: 1 × 12, 2 × 6, 3 × 4.

A multiple of a number is the result of multiplying it by a positive integer. The first five multiples of 7 are 7, 14, 21, 28, and 35.

A prime number is a number greater than 1 that has exactly two factors: 1 and itself. The primes up to 30 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Note that 1 is not prime and 2 is the only even prime.

A prime factor tree breaks a number down into a product of its prime factors. Every integer greater than 1 has a unique prime factorisation.

Key Formulas

Step-by-Step Method

Finding All Factors

1. Start with 1 and the number itself — this is your first factor pair.
2. Test 2, 3, 4, ... checking if each divides exactly into the number.
3. Stop when your test number meets or exceeds the factor you are pairing it with.
4. List all factors in order.

Drawing a Prime Factor Tree

1. Write the number at the top.
2. Split it into any two factors.
3. Circle any prime factors (they are the branches that stop).
4. Keep splitting non-prime factors until every branch ends in a prime.
5. Write the result as a product of primes using index notation.

Worked Example 1 — Foundation Level

Question: List all the factors of 36.

Working:

Step 1 — Start: 1 × 36. Step 2 — 2 × 18. Step 3 — 3 × 12. Step 4 — 4 × 9. Step 5 — 6 × 6.

Since 6 × 6 gives a repeated factor, we stop.

Answer: 1, 2, 3, 4, 6, 9, 12, 18, 36

Worked Example 2 — Higher Level

Question: Write 360 as a product of its prime factors. Give your answer in index notation.

Working:

Step 1 — Factor tree: 360 = 2 × 180. 180 = 2 × 90. 90 = 2 × 45. 45 = 3 × 15. 15 = 3 × 5.

Step 2 — Collect the primes: 2 × 2 × 2 × 3 × 3 × 5.

Step 3 — Write in index notation: 2³ × 3² × 5.

Answer: 360 = 2³ × 3² × 5

Worked Example 3 — Exam Style

Question: Explain why 51 is not a prime number.

Working:

Step 1 — Check divisibility. 51 ÷ 3 = 17 (no remainder).

Step 2 — Since 51 = 3 × 17, it has factors other than 1 and itself.

Answer: 51 is not prime because it is divisible by 3 (51 = 3 × 17).

Common Mistakes

• Thinking 1 is a prime number. It is not — a prime must have exactly two distinct factors, and 1 only has one factor.
• Missing factor pairs. Work systematically from 1 upwards to ensure you find every pair. Stop when the factors start repeating.
• Not fully breaking down the factor tree. Every branch must end at a prime. If you stop at 4, 6, 9, or any composite number, the factorisation is incomplete.

Exam Tips

• Memorise the prime numbers up to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
• Use divisibility rules: a number is divisible by 3 if its digit sum is divisible by 3.
• Always present your final answer in index notation — examiners expect this.

Practice Questions

Q1 (Foundation): List all the factor pairs of 48.

Q2 (Foundation): Write down the first six multiples of 9.

Q3 (Higher): Write 504 as a product of its prime factors in index notation.

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Related Topics

• Prime Factorisation, HCF and LCM
• Indices and Index Laws
• Equivalent Fractions and Simplifying

Summary

• Factors divide exactly into a number; list them systematically in pairs.
• Multiples are found by multiplying the number by 1, 2, 3, and so on.
• A prime number has exactly two factors: 1 and itself. The number 1 is not prime.
• Use a factor tree to express any number as a product of primes in index notation.