GCSE Maths Practice: factors-and-multiples

Question 7 of 10

This GCSE Maths question tests your understanding of the Least Common Multiple (LCM) — the smallest number that appears in both times tables.

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\( \begin{array}{l}\text{What is the least common multiple (LCM) of 6 and 8?}\end{array} \)

Choose one option:

List a few multiples of each number and compare them carefully. The smallest one they both share is the LCM. Double-check by dividing both numbers into it — there should be no remainder.

Understanding the Least Common Multiple (LCM)

The Least Common Multiple (LCM) of two numbers is the smallest number that appears in both of their times tables. It is one of the key skills tested in GCSE Maths because it helps when adding or subtracting fractions, solving ratio problems, or aligning repeating events in real-life contexts.

For example, if two machines operate on different time cycles, the LCM tells us when they will next finish a cycle together.

Step-by-Step Method

  1. Write out several multiples of each number.
  2. Compare both lists to see which numbers appear in both.
  3. The smallest of those shared numbers is the LCM.

Worked Examples (Different Values)

  • Example 1: Find the LCM of 3 and 4.
    3 → 3, 6, 9, 12, 15...
    4 → 4, 8, 12, 16, 20... → LCM = 12.
  • Example 2: Find the LCM of 5 and 6.
    5 → 5, 10, 15, 20, 25, 30...
    6 → 6, 12, 18, 24, 30, 36... → LCM = 30.
  • Example 3: Find the LCM of 7 and 9.
    7 → 7, 14, 21, 28, 35, 42, 49, 56, 63...
    9 → 9, 18, 27, 36, 45, 54, 63... → LCM = 63.

Alternative Method: Prime Factorisation

Prime factorisation is a quick and accurate way to find the LCM, especially for larger numbers.

  1. Write each number as a product of its prime factors.
  2. Include every prime factor that appears in either number, using the highest power of each.
  3. Multiply these together to find the LCM.

Example: 6 = 2 × 3, 8 = 2³ → use 2³ × 3 = 24 → LCM = 24.

Common Mistakes

  • Mixing up HCF and LCM: The HCF (Highest Common Factor) is the largest shared divisor, while the LCM is the smallest shared multiple.
  • Stopping too soon: Some learners list too few multiples and miss the first shared value.
  • Using factors instead of multiples: Factors are smaller or equal to the original number, while multiples are larger or equal.

Real-Life Applications

LCM problems appear frequently in everyday life:

  • Scheduling: If one bus leaves every 6 minutes and another every 8 minutes, they leave together every LCM(6, 8) = 24 minutes.
  • Cooking and Production: When two machines or recipes repeat cycles at different intervals, the LCM shows when both cycles align again.
  • Fractions: The LCM is used to find a common denominator when adding or subtracting fractions like 1/6 and 1/8.

Frequently Asked Questions

Q1: What happens if one number divides exactly into the other?
A: Then the LCM is the larger number. For example, LCM(5, 10) = 10.

Q2: Can the LCM be smaller than both numbers?
A: No, it is always equal to or greater than the largest number.

Q3: Why is it called 'least' common multiple?
A: Because it’s the smallest number shared by both lists of multiples, even though there are many larger ones too.

GCSE Study Tip

To save time, look for the smallest common multiple by scanning the multiplication tables rather than listing too many terms. For higher numbers, use prime factorisation — it’s faster and avoids missing the smallest shared value.

Summary

The Least Common Multiple (LCM) is the smallest number that both numbers share in their times tables. In this question, the LCM of 6 and 8 is 24. Understanding how to find LCMs helps with fractions, ratios, and real-world problems like scheduling or packaging. Mastering this process ensures accuracy in GCSE Maths exams and builds strong number sense.