GCSE Maths Practice: factors-and-multiples

Question 10 of 10

This GCSE Maths question focuses on identifying factors — numbers that divide exactly into another number. Mastering this skill helps simplify fractions and solve grouping problems.

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\( \begin{array}{l}\text{Which of these is a factor of 45?}\end{array} \)

Choose one option:

Divide the given number by each option. If the result is a whole number, it’s a factor. Checking through divisibility rules saves time in exams.

Understanding Factors

A factor is a number that divides another number exactly with no remainder. In GCSE Maths, factors help break down numbers into their building blocks, making it easier to simplify fractions, find common factors, and understand multiplication relationships. For example, since 45 can be written as 3 × 15, both 3 and 15 are factors of 45.

How to Identify Factors

  1. Start by dividing the number by smaller integers beginning with 1.
  2. If the result is a whole number, that divisor is a factor.
  3. Continue until you reach the square root of the number — after that, the factors repeat in pairs.

For instance, 45 ÷ 1 = 45, 45 ÷ 3 = 15, 45 ÷ 5 = 9, 45 ÷ 9 = 5, 45 ÷ 15 = 3. So, the factors of 45 are 1, 3, 5, 9, 15, and 45.

Worked Examples (Different Values)

  • Example 1: Find the factors of 28.
    1, 2, 4, 7, 14, 28 → all divide exactly into 28.
  • Example 2: Find the factors of 36.
    1, 2, 3, 4, 6, 9, 12, 18, 36.
  • Example 3: Find the factors of 60.
    1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

Quick Divisibility Rules

These mental checks help you test whether a number is a factor quickly:

  • Divisible by 2: Ends in 0, 2, 4, 6, or 8.
  • Divisible by 3: Sum of digits divisible by 3 (e.g. 4 + 5 = 9 → divisible).
  • Divisible by 5: Ends in 0 or 5.
  • Divisible by 9: Sum of digits divisible by 9 (e.g. 4 + 5 = 9 → divisible).

Using these, we can see that 45 is divisible by both 3, 5, 9, and 15, confirming they are all factors.

Difference Between Factors and Multiples

  • Factors divide into a number exactly (they are smaller or equal).
  • Multiples are the results of multiplying a number by integers (they are larger or equal).

Example: Factors of 45 = 1, 3, 5, 9, 15, 45; Multiples of 15 = 15, 30, 45, 60...

Common Mistakes

  • Forgetting factor pairs: Each factor below the square root pairs with one above it.
  • Mixing up factors and multiples: Remember: factors fit inside the number; multiples extend beyond it.
  • Leaving out 1 or the number itself: These are always factors.

Real-Life Applications

Recognising factors helps solve everyday and mathematical problems:

  • Sharing equally: If 45 sweets must be shared evenly, possible group sizes are factors of 45 (e.g. 3, 5, 9, or 15 people).
  • Fractions: Simplify 45⁄60 by dividing numerator and denominator by their HCF (15) → 3⁄4.
  • Area design: To create rectangles with 45 tiles, dimensions come from factor pairs such as 3 × 15 or 5 × 9.

Frequently Asked Questions

Q1: How can I check quickly if a number is a factor?
A: Divide — if the result is a whole number, it’s a factor.

Q2: Can negative numbers be factors?
A: Yes. For example, −3 and −15 are also factors of 45, but in GCSE Maths we usually use positive ones.

Q3: How many factors does 45 have?
A: Six — 1, 3, 5, 9, 15, and 45.

GCSE Study Tip

When finding factors, list them in pairs to avoid repetition. Use divisibility rules to save time. Practising factor lists regularly improves accuracy in HCF, LCM, and ratio questions.

Summary

A factor divides another number exactly with no remainder. In this question, 15 is a factor of 45 because 45 ÷ 15 = 3. Understanding factors builds essential foundations for simplifying fractions, solving ratio problems, and recognising number patterns in GCSE Maths.