GCSE Maths Practice: factors-and-multiples

Question 5 of 10

This GCSE Maths question focuses on finding the Least Common Multiple (LCM) of two numbers — a key concept used in fractions, ratios, and scheduling problems.

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\( \begin{array}{l}\text{What is the least common multiple (LCM) of 2 and 5?}\end{array} \)

Choose one option:

LCM problems often appear with fractions and ratio questions. List multiples clearly and pick the smallest common one to ensure accuracy.

Understanding the Least Common Multiple (LCM)

The Least Common Multiple (LCM) is the smallest number that two or more numbers divide into exactly. In GCSE Maths, the LCM is often used in fraction problems, ratio questions, and finding common time intervals. It tells us when two repeating patterns coincide — for example, when two traffic lights flash at the same time again.

Step-by-Step Method

  1. Write the first few multiples of each number.
  2. Compare both lists to find the smallest number that appears in both.
  3. That smallest shared number is the LCM.

Alternatively, you can use prime factorisation to find the LCM more efficiently, especially for larger numbers.

Worked Examples (Different Values)

  • Example 1: Find the LCM of 3 and 5.
    Multiples of 3: 3, 6, 9, 12, 15, 18...
    Multiples of 5: 5, 10, 15, 20, 25...
    Common multiples: 15, 30, 45... → LCM = 15.
  • Example 2: Find the LCM of 4 and 6.
    Multiples of 4: 4, 8, 12, 16, 20...
    Multiples of 6: 6, 12, 18, 24... → LCM = 12.
  • Example 3: Find the LCM of 8 and 10.
    Multiples of 8: 8, 16, 24, 32, 40...
    Multiples of 10: 10, 20, 30, 40, 50... → LCM = 40.

Alternative Method: Prime Factorisation

  1. Write both numbers as products of prime factors.
  2. Take each unique prime factor the highest number of times it occurs.
  3. Multiply them together to get the LCM.

Example: 4 = 2², 6 = 2 × 3 → LCM = 2² × 3 = 12.

Common Mistakes

  • Choosing the greatest instead of least: The HCF is the largest common factor; the LCM is the smallest shared multiple — don’t confuse the two.
  • Stopping too soon: Always list enough multiples to find the first common one.
  • Misusing factors: Remember that multiples are larger than the original number, not smaller.

Real-Life Applications

LCM appears in many everyday situations. It’s used to coordinate events, schedules, and cycles:

  • Timetables: If buses leave every 5 minutes and trains every 8 minutes, they’ll both leave together every LCM(5, 8) = 40 minutes.
  • Production cycles: In factories, if two machines complete cycles every 12 and 15 seconds, they finish together every 60 seconds.
  • Fractions: To add or subtract fractions, the LCM helps find a common denominator.

Frequently Asked Questions

Q1: Can two numbers have more than one common multiple?
A: Yes, multiples go on infinitely, but the smallest one is the LCM.

Q2: Can the LCM ever be one of the numbers?
A: Yes, when one number divides exactly into the other. For example, LCM(4, 8) = 8.

Q3: How is LCM different from HCF?

A: The HCF is the biggest shared factor (divides into both), while the LCM is the smallest shared multiple (both divide into it).

GCSE Study Tip

When revising, practise listing multiples for pairs of numbers quickly — it helps with both LCM and HCF questions. For higher numbers, use prime factorisation to save time and reduce mistakes.

Summary

The Least Common Multiple (LCM) is the smallest number that two or more numbers share in their times tables. In this question, the LCM of 2 and 5 is 10. Understanding this method is essential for fraction operations, timetabling, and ratio-based problem solving in GCSE Maths.