GCSE Maths Practice: factors-and-multiples

Question 10 of 10

This problem applies the concept of LCM to a real-life scheduling situation involving three repeating time intervals.

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\( \begin{array}{l}\text{Bus A arrives every 15 min, Bus B every 25 min,}\\\text{and Bus C every 30 min. All leave at 9:00 a.m.}\text{ When will they next all arrive together?}\end{array} \)

Choose one option:

Identify the repeating time intervals, find the least common multiple (LCM), and convert the result to hours and minutes.

Understanding LCM Through Real-Life Problems

The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. At GCSE Higher level, this concept frequently appears in word problems involving events that repeat at different intervals — such as bus timetables, flashing lights, or machinery cycles. Recognising when all cycles coincide is a practical application of LCM.

Worked Example: Bus Timetables

Bus A arrives every 15 minutes, Bus B every 25 minutes, and Bus C every 30 minutes. All three buses arrive together at 9:00 a.m. When will they next arrive together?

To find out, calculate the least common multiple of 15, 25, and 30.

15 = 3 × 5
25 = 5 × 5 = 5^2
30 = 2 × 3 × 5

Identify all unique prime factors: 2, 3, and 5. Take the highest powers of each:

LCM = 2 × 3 × 5^2 = 2 × 3 × 25 = 150.

The buses will meet again in 150 minutes, which is 2 hours and 30 minutes after 9:00 a.m., so at 11:30 a.m.

Alternative Example (Different Context)

Three lights flash every 18, 24, and 30 seconds. They flash together at 8:00:00. Find the next time they flash together.

18 = 2 × 3^2
24 = 2^3 × 3
30 = 2 × 3 × 5
LCM = 2^3 × 3^2 × 5 = 360 seconds = 6 minutes.

The lights flash together every 6 minutes.

Common Mistakes

  • Confusing LCM with GCD — LCM uses the highest powers of each prime factor, while GCD uses the lowest.
  • Listing multiples too short — stop only when you find the first one that appears in all lists.
  • Forgetting to convert the result into the correct units (e.g., minutes to hours).

Real-Life Applications

LCM appears in many practical settings:

  • Scheduling: Coordinating repeating events like buses, shifts, or alarms.
  • Engineering: Synchronising machine rotations or gear teeth cycles.
  • Science: Calculating when repeating waves or frequencies align.
  • Maths & Algebra: Finding a common denominator when adding fractions.

FAQ

Q: What’s the difference between LCM and GCD?
A: The LCM is the smallest shared multiple, while the GCD is the largest shared factor.

Q: Why use prime factorisation?
A: It ensures accuracy with larger numbers and avoids missing hidden common multiples.

Q: How can I check my LCM?

A: Divide the LCM by each original number — the result should always be a whole number.

Study Tip

When facing a word problem, rewrite it numerically first. Identify which events repeat and find the LCM to know when they coincide. Practise converting between minutes and hours to make your answers clear and practical.

Summary

The LCM is more than a number trick — it represents the point when repeating patterns align. Understanding it helps solve real-life problems and lays the foundation for working with time, cycles, and fractions in GCSE Higher Maths.