GCSE Maths Practice: factors-and-multiples

Question 7 of 10

This question applies LCM to a real-world scheduling context where four repeating shifts must align again.

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\( \begin{array}{l}\text{Four cleaning teams start work together at 8:00 a.m.}\\\text{Their shifts repeat every 6, 8, 9, and 12 hours.}\\\text{After how many hours will they next start together?}\end{array} \)

Choose one option:

Find the least common multiple (LCM) of all four intervals and convert the result into days if necessary.

Understanding LCM in Scheduling Problems

The least common multiple (LCM) represents the first time that two or more repeating events coincide. In Higher GCSE Maths, LCM appears in word problems where multiple activities or schedules repeat at different intervals. Mastering this idea helps you model real situations such as maintenance cycles, staff shifts, and repeating tasks.

Worked Example: Cleaning Rotas

Four cleaning teams start work together at 8:00 a.m. Team A repeats every 6 hours, Team B every 8 hours, Team C every 9 hours, and Team D every 12 hours. When will they all start a shift at the same time again?

We need the least common multiple of 6, 8, 9, and 12.

6 = 2 × 3
8 = 2 × 2 × 2 = 2^3
9 = 3 × 3 = 3^2
12 = 2^2 × 3

Take the highest powers of each prime: 2³ and 3².

LCM = 2³ × 3² = 8 × 9 = 72.

The teams will all begin a new shift together every 72 hours, which equals 3 days.

Alternative Example (Different Context)

Four students charge their tablets every 6, 8, 9, and 12 hours. The least common multiple of 6, 8, 9, and 12 is again 72, meaning they all plug in at the same time every 72 hours.

Why This Method Works

Prime factorisation ensures no multiple is missed. By taking the highest power of each prime, we include all factors needed for each schedule to fit evenly into the total time.

Common Mistakes

  • Using the lowest powers instead of the highest (that gives the GCD, not LCM).
  • Mixing units — forgetting that 72 hours equals 3 days.
  • Stopping at a number that fits only some of the schedules.

Real-Life Applications

  • Operations management: Aligning staff schedules, production runs, or maintenance windows.
  • Transport planning: Coordinating buses, trains, or aircraft cycles.
  • Computer science: Synchronising repeating data refresh or backup intervals.
  • Music and design: Matching repeating patterns or beats that overlap at specific intervals.

FAQ

Q: Why use prime factorisation instead of listing multiples?
A: It’s faster and avoids missing hidden overlaps, especially with several numbers.

Q: What if two of the intervals share the same multiple?
A: The LCM still includes both — that’s why we take the highest powers.

Q: Can we just multiply all the numbers?
A: No. Multiplying gives a common multiple, but not the least one.

Study Tip

When working with several intervals, list each prime factorisation vertically so you can clearly see the highest powers. Then multiply only the unique primes at their maximum powers. Always convert your final result into meaningful units such as minutes, hours, or days.

Summary

The LCM is a powerful concept for solving multi-event timing problems. By calculating when different cycles align, you can model real-world scenarios involving scheduling, engineering, or repeating sequences. It’s a key Higher GCSE skill connecting number theory to practical reasoning.