GCSE Maths Practice: factors-and-multiples

Question 6 of 10

This GCSE Maths question explores how to find the Least Common Multiple (LCM) of three numbers. It’s a useful technique for combining cycles, aligning events, and working with fractions that have more than two denominators.

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\( \begin{array}{l}\text{What is the least common multiple (LCM) of 3, 4, and 5?}\end{array} \)

Choose one option:

When three numbers are involved, first find the LCM of any two and then combine it with the third. Prime factorisation makes this faster and avoids listing too many multiples.

Understanding the Least Common Multiple (LCM) of Three Numbers

The Least Common Multiple (LCM) is the smallest number that all given numbers divide into exactly. When you work with three numbers, the same principle applies — it’s the first number that appears in all three times tables. Learning to find the LCM of three numbers helps with GCSE Maths problems involving fractions, repeating patterns, or simultaneous cycles.

Step-by-Step Method

  1. List multiples of two of the numbers. Find their LCM first.
  2. Use that result to find the LCM with the third number.
  3. The final answer is the smallest number divisible by all three.

This method avoids listing very long sets of multiples and makes calculation faster.

Worked Examples (Different Values)

  • Example 1: LCM of 3, 4, and 5.
    LCM(3, 4) = 12. Then LCM(12, 5) = 60 → LCM = 60.
  • Example 2: LCM of 2, 3, and 6.
    LCM(2, 3) = 6, LCM(6, 6) = 6 → LCM = 6.
  • Example 3: LCM of 4, 6, and 10.
    LCM(4, 6) = 12, LCM(12, 10) = 60 → LCM = 60.
  • Example 4: LCM of 5, 7, and 9.
    LCM(5, 7) = 35, LCM(35, 9) = 315 → LCM = 315.

Prime Factorisation Method

Using prime factors makes finding the LCM of three numbers much quicker:

  1. Write each number as a product of primes.
  2. Include every prime factor that appears in any of the numbers, using the highest power found.
  3. Multiply them all together to get the LCM.

Example: 3 = 3, 4 = 2², 5 = 5. LCM = 2² × 3 × 5 = 60.

Common Mistakes

  • Stopping too early: Some students forget to include the third number after finding the LCM of the first two.
  • Mixing up HCF and LCM: The LCM is the smallest common multiple, while the HCF is the largest shared factor.
  • Forgetting the highest powers: When using prime factorisation, always take the highest power of each prime — otherwise, the result is too small.

Real-Life Applications

The concept of the LCM for three numbers appears in real-world scheduling, science, and engineering:

  • Timetables: If buses, trains, and ferries run every 3, 4, and 5 minutes, they all meet at the same time every 60 minutes.
  • Science experiments: Timing cycles that repeat at different intervals use the LCM to find when they coincide.
  • Fractions and ratios: When combining three fractions, the LCM of the denominators provides the common base.

Frequently Asked Questions

Q1: Can the LCM ever equal one of the numbers?
A: Yes, when one number is a multiple of all the others. For example, LCM(4, 8, 16) = 16.

Q2: What is the difference between the LCM of two and three numbers?
A: None — the principle is the same; you just extend the method to include a third value.

Q3: Can the LCM of three numbers be odd?
A: Yes, if all the numbers are odd, such as LCM(3, 5, 7) = 105.

GCSE Study Tip

To avoid listing long sequences, use prime factorisation — it’s the fastest way to find the LCM of two or more numbers. Remember: take each prime factor the highest number of times it appears in any of the numbers.

Summary

The Least Common Multiple (LCM) of several numbers is the smallest number that all divide into exactly. To find it, either list multiples or use prime factorisation. In this example, LCM(3, 4, 5) = 60. Understanding this method helps with fractions, repeating cycles, and ratio problems in GCSE Maths.