GCSE Maths Practice: factors-and-multiples

Question 2 of 10

This Higher-level question applies the concept of the least common multiple (LCM) to a practical timing problem involving two machines operating on repeating cycles.

‹ ›

\( \begin{array}{l}\text{Machine A completes a cycle every 18 min,}\\\text{and Machine B every 24 min. Both start at 9:00 a.m.}\\\text{When will both start together again?}\end{array} \)

Choose one option:

Find the LCM of the two time intervals, then convert the result from minutes to hours and minutes.

Using LCM in Scheduling and Timing Problems

The least common multiple (LCM) represents the smallest time or quantity at which two or more repeating cycles coincide. In Higher GCSE Maths, LCM questions often appear in the form of word problems that test whether students can recognise how periodic events align in time or sequence.

Worked Example: Machines in a Workshop

Machine A completes a production cycle every 18 minutes, and Machine B completes one every 24 minutes. Both start their first cycle at 9:00 a.m. When will both machines start a cycle at the same time again?

This situation requires the least common multiple of 18 and 24, since we are looking for when their cycles overlap.

18 = 2 × 3 × 3 = 2 × 3^2
24 = 2 × 2 × 2 × 3 = 2^3 × 3

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

LCM = 2³ × 3² = 8 × 9 = 72.

The cycles align every 72 minutes. That means both machines will start together again at 10:12 a.m.

Alternative Example (Different Context)

Two traffic lights change at intervals of 18 and 24 seconds. To find when they will next turn green at the same time, you again calculate the LCM of 18 and 24. The result, 72 seconds, shows that both lights turn green together every 1 minute and 12 seconds.

Common Mistakes

  • Confusing LCM with GCD – remember, LCM uses the highest powers of all prime factors.
  • Forgetting to convert the answer back into practical units (e.g., hours and minutes).
  • Stopping at a smaller multiple that isn’t shared by both numbers.
  • Not recognising that both times or cycles must restart together, not just finish.

Real-Life Applications

The concept of LCM extends far beyond pure maths. It’s used in:

  • Manufacturing: Scheduling machines or robots to sync in production cycles.
  • Music: Determining when two rhythms with different beat patterns align.
  • Computer Science: Synchronising repeating processes or data refresh rates.
  • Engineering: Calculating when gears with different rotations align.

Quick FAQ

Q: Can the LCM ever be smaller than both numbers?
A: No. The LCM is always equal to or greater than the largest number.

Q: Why are highest powers used?
A: Because every number must divide evenly into the LCM, which requires keeping all prime factors to their largest powers.

Q: How do I check my result?
A: Divide your LCM by both original numbers — both results must be whole numbers.

Study Tip

When solving LCM word problems, always underline the time or cycle lengths in the text, then extract the numbers before calculating. This prevents confusion and ensures your mathematical setup matches the real situation.

Summary

LCM is not just a mechanical process; it’s a logical tool to coordinate repeating events. Whether synchronising machines, signals, or schedules, understanding how to calculate and interpret the least common multiple is a core skill for success in GCSE Higher Maths.