GCSE Maths Practice: factors-and-multiples

Question 3 of 10

This problem applies the least common multiple (LCM) to a real-world event scheduling scenario involving flashing lights at a festival.

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\( \begin{array}{l}\text{Three light displays flash every 7, 8, and 12 seconds.}\\\text{They all flash together at 6:00:00 p.m.}\text{ When will they next flash together?}\end{array} \)

Choose one option:

Find the least common multiple (LCM) of the three flashing intervals and convert the total into minutes and seconds for a practical answer.

Finding When Repeating Events Align

The least common multiple (LCM) describes the first moment when two or more repeating cycles coincide. In GCSE Higher Maths, it’s often used in time-based problems — from traffic lights and machinery to festivals and timing devices. These questions test your understanding of periodicity and prime factorisation.

Worked Example: Light Displays at a Festival

Three light displays flash at different intervals: one every 7 seconds, one every 8 seconds, and one every 12 seconds. All three flash together at 6:00:00 p.m. When will they flash together again?

We need the least common multiple of 7, 8, and 12.

7 = 7
8 = 2^3
12 = 2^2 × 3

Identify the highest powers of all primes that appear:

2^3, 3^1, and 7^1

Now multiply them together:

LCM = 2^3 × 3 × 7 = 8 × 21 = 168.

So, all three light displays will flash together again after 168 seconds, which is equivalent to 2 minutes and 48 seconds.

Alternative Example (Different Context)

Three rotating warning signals turn every 7, 8, and 12 minutes. The LCM of 7, 8, and 12 is also 168, meaning they all align every 168 minutes, or 2 hours and 48 minutes. This is the first time they complete a full synchronised cycle.

Understanding Why LCM Works

The LCM ensures all events complete an exact number of cycles before repeating together. Each interval divides evenly into the total LCM time, so all events ‘reset’ simultaneously at that moment.

Common Mistakes

  • Mixing up GCD and LCM — GCD finds the largest shared factor, not the smallest shared multiple.
  • Multiplying the numbers directly without removing duplicates in prime factors.
  • Forgetting to convert seconds into minutes or hours after calculation.

Real-Life Applications

  • Event management: Coordinating sound, light, and pyrotechnic systems.
  • Technology: Synchronising blinking LEDs or processors that operate at different cycles.
  • Engineering: Timing multiple rotating parts or mechanical events.
  • Science: Predicting when cyclical phenomena (waves, orbits) coincide.

FAQ

Q: What if two of the cycles share a factor?
A: You still take the highest power of each prime — shared primes don’t need to be repeated.

Q: Why is 168 the smallest time that works?
A: Because it’s the first value that’s divisible by 7, 8, and 12 simultaneously.

Q: How can I verify my answer?
A: Divide 168 by each number. The results (24, 21, 14) are all whole, confirming divisibility.

Study Tip

In time-based LCM problems, always check your units and convert large results into minutes or hours for a practical answer. Understanding the pattern helps you quickly recognise real-life LCM contexts in exams.

Summary

LCM problems demonstrate how mathematics predicts when multiple repeating events will synchronise. Whether timing festival lights, factory cycles, or transport schedules, finding the least common multiple is an essential Higher GCSE skill that bridges number theory and practical reasoning.