GCSE Maths Practice: factors-and-multiples

Question 9 of 10

This GCSE Maths question focuses on identifying factors — numbers that divide another number exactly without leaving a remainder. Understanding factors supports work on fractions, ratios, and number patterns.

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\( \begin{array}{l}\text{Which of these is a factor of 18?}\end{array} \)

Choose one option:

To identify factors, divide the given number by each option. If the result is a whole number, that option is a factor. Always check by multiplication to confirm.

Understanding Factors

A factor is a number that divides another number exactly, leaving no remainder. Factors show how a number can be built by multiplication. For example, 18 can be written as 3 × 6, so both 3 and 6 are factors of 18. Learning to identify factors helps with simplifying fractions, solving ratio problems, and understanding divisibility rules in GCSE Maths.

Step-by-Step Method

  1. Start with 1 and the number itself — these are always factors.
  2. Test smaller numbers by dividing the given number by them.
  3. If the result is a whole number, the divisor is a factor.
  4. Stop once the numbers begin repeating — factors always come in pairs.

Worked Examples (Different Values)

  • Example 1: Find the factors of 12.
    1, 2, 3, 4, 6, 12 → factors come in pairs (1×12, 2×6, 3×4).
  • Example 2: Find the factors of 20.
    1, 2, 4, 5, 10, 20 → 4 × 5 = 20.
  • Example 3: Find the factors of 15.
    1, 3, 5, 15 → all divide exactly into 15.

How to Check Quickly

You can often recognise factors without full division by using simple patterns:

  • Even numbers: All even numbers are divisible by 2.
  • Ending in 0 or 5: Divisible by 5.
  • Digit sum divisible by 3: Then the number is divisible by 3.
  • Divisible by 10: Must end in 0.

For instance, 18 is even (so divisible by 2) and its digits (1 + 8 = 9) are divisible by 3, meaning both 2 and 3 — and hence 6 — are factors of 18.

Difference Between Factors and Multiples

Students often mix up these two terms:

  • Factors are numbers that fit inside another number (they divide it).
  • Multiples are numbers that go beyond by multiplying (they’re in the times table of that number).

Example: Factors of 18 = 1, 2, 3, 6, 9, 18; Multiples of 6 = 6, 12, 18, 24, 30…

Common Mistakes

  • Confusing factors with multiples: Always check whether the number divides exactly — if not, it’s not a factor.
  • Forgetting factor pairs: Every factor below the square root has a matching one above it.
  • Ignoring 1: 1 is a factor of every integer.

Real-Life Applications

Understanding factors is useful in many practical situations:

  • Packaging: To divide 18 items equally into boxes with no leftovers, box sizes must be factors of 18 (1, 2, 3, 6, 9, or 18).
  • Fractions: Simplifying 18⁄24 by dividing top and bottom by 6 → 3⁄4.
  • Area and design: When arranging tiles or shapes, factor pairs help you find all possible rectangle dimensions.

Frequently Asked Questions

Q1: Can a number be its own factor?
A: Yes, every number is a factor of itself because it divides exactly once.

Q2: How many factors does 18 have?
A: Six — they are 1, 2, 3, 6, 9, and 18.

Q3: What is the smallest factor of any number?
A: 1, since 1 divides every number exactly.

GCSE Study Tip

To find factors quickly, test divisibility rules and stop once you reach the square root of the number. For larger values, use prime factorisation — it guarantees you won’t miss any factors.

Summary

A factor divides another number exactly with no remainder. In this question, 6 is a factor of 18 because 18 ÷ 6 = 3. Recognising and listing factors is an essential foundation for many GCSE Maths topics, including fractions, ratios, and algebraic simplification.