GCSE Maths Practice: factors-and-multiples

Question 1 of 10

This GCSE Maths question focuses on identifying factors — numbers that divide exactly into another number. Understanding factors helps with simplifying fractions, ratios, and multiplication problems.

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\( \begin{array}{l}\text{Is 25 a factor of 100?}\end{array} \)

Choose one option:

To test if a number is a factor, divide it into the larger number. If there’s no remainder, it is a factor. If there is, it isn’t.

Understanding Factors and Multiples

In GCSE Maths, a factor is a number that divides another number exactly, leaving no remainder. For example, 4 is a factor of 12 because 12 ÷ 4 = 3 with no remainder. Factors show how a number can be built by multiplication, while multiples show how it can be extended by repeated addition or multiplication.

The relationship between factors and multiples is key to understanding number structure, simplifying fractions, and solving problems involving divisibility and prime numbers.

How to Test if a Number is a Factor

  1. Take the number you want to test as a potential factor.
  2. Divide the larger number by it.
  3. If the result is a whole number (integer), it divides exactly — meaning it is a factor.
  4. If there’s a remainder or a decimal result, it is not a factor.

Worked Examples (Different Values)

  • Example 1: Is 5 a factor of 45?
    45 ÷ 5 = 9 → divides exactly → Yes.
  • Example 2: Is 6 a factor of 50?
    50 ÷ 6 = 8.33 → not exact → No.
  • Example 3: Is 8 a factor of 64?
    64 ÷ 8 = 8 → divides exactly → Yes.

Difference Between Factors and Multiples

  • Factors are numbers that fit inside another number (they divide it exactly).
  • Multiples are the results when a number is multiplied by 1, 2, 3, and so on.

For example, the factors of 20 are 1, 2, 4, 5, 10, 20, while the multiples of 4 are 4, 8, 12, 16, 20, 24, etc.

Common Mistakes

  • Confusing factors with multiples: Remember that factors are smaller or equal to the number, while multiples are larger.
  • Forgetting to check divisibility: Use division or multiplication to test accurately — guessing can cause small but costly exam errors.
  • Including 0 as a factor: Zero is never a factor because you cannot divide by zero.

Real-Life Applications

Understanding factors helps simplify fractions, divide quantities fairly, and work out arrangements in grids, patterns, or packaging. For example, if 60 pencils need to be packed evenly into boxes, factors help determine how many boxes can be used. Similarly, in computer science or data grouping, factors are used to balance and organise datasets efficiently.

Frequently Asked Questions

Q1: Is 1 always a factor?
A: Yes. Every number is divisible by 1 exactly once.

Q2: Is the number itself always a factor?
A: Yes. Any whole number divides by itself to give 1.

Q3: Can negative numbers be factors?

A: Yes, in theory. For example, −4 × −3 = 12, so both −4 and −3 are also factors of 12, though we usually use positive ones in GCSE Maths.

GCSE Study Tip

Learn to list factors systematically. Start with 1 and the number itself, then test every integer in between. Stop when the pairs start repeating. This technique also helps when finding the Highest Common Factor (HCF) of two numbers.

Summary

To decide if one number is a factor of another, divide and check for no remainder. This principle is at the heart of many GCSE Maths topics, including simplification, ratio, and prime factorisation. Factors show the building blocks of numbers — once you master them, larger concepts like multiples, HCF, and LCM become far easier to handle.