GCSE Maths Practice: factors-and-multiples

Question 1 of 10

This question applies the least common multiple (LCM) to a physics-style context about rotating wheels or gears.

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\( \begin{array}{l}\text{Wheel A makes a full rotation every 4 s,}\\\text{Wheel B every 10 s, and Wheel C every 15 s.}\\\text{After how many seconds will they align again?}\end{array} \)

Choose one option:

Convert the problem to rotation intervals, find the least common multiple (LCM), and express the answer in seconds.

LCM and Rotational Motion

The least common multiple (LCM) identifies when repeating events coincide. In physics and engineering problems, LCM helps determine when rotating parts, gears, or signals align again after different cycles. Understanding this concept is crucial for synchronisation in both mechanical and digital systems.

Worked Example: Rotating Wheels

Three wheels start turning at the same time. Wheel A completes one revolution every 4 seconds, Wheel B every 10 seconds, and Wheel C every 15 seconds. After how many seconds will all three be in their starting position together?

We calculate the least common multiple of 4, 10, and 15, because we need to know when their rotations match again.

4 = 2 × 2 = 2^2
10 = 2 × 5
15 = 3 × 5
LCM = 2^2 × 3 × 5 = 4 × 3 × 5 = 60.

The wheels will align again every 60 seconds.

Alternative Example (Different Context)

Three traffic beacons flash every 4, 6, and 10 seconds. All flash together at 12:00:00. The LCM of 4, 6, and 10 is 60, meaning they will next flash together at 12:01:00.

Why LCM Works

The LCM ensures that each cycle finishes a whole number of times within the same total time. If an event repeats every 4 seconds, after 60 seconds it has completed 15 full cycles (60 ÷ 4 = 15). For the others: 6 cycles for the 10-second event, and 4 for the 15-second one. All three meet perfectly after 60 seconds.

Common Mistakes

  • Confusing GCD and LCM — GCD is for dividing things into equal parts, LCM is for finding when things coincide.
  • Leaving the answer as prime factors instead of the actual number.
  • Forgetting that the LCM must be a multiple of all given numbers, not just some.

Real-Life Applications

  • Engineering: Synchronising gears, pistons, or conveyor belts that operate at different speeds.
  • Sound waves: Finding when frequencies overlap to form beats.
  • Computing: Determining refresh or signal timing for synchronised systems.
  • Music: Finding when repeating rhythms line up in a composition.

FAQ

Q: How can I verify my LCM quickly?
A: Divide your answer by each original number. All results should be whole numbers.

Q: What happens if one number is a factor of another?
A: Then the LCM equals the larger number.

Q: Can I use multiples instead of primes?
A: Yes — listing multiples works for small numbers, but prime factorisation is faster and more accurate.

Study Tip

When facing LCM problems, visualise each event as a cycle. For time-based questions, note that the LCM gives the exact time when all cycles ‘reset’ together. Always check your result makes sense in real-world units — seconds, minutes, or hours.

Summary

In this problem, the LCM shows when rotating or repeating events line up again. This principle underpins many areas of maths, science, and engineering. Mastering it will help you move confidently into advanced problem-solving across multiple subjects.