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LCM and HCF

FoundationHigherAQAEdexcelOCR

Get to grips with LCM and HCF using these Foundation and Higher GCSE Maths practice questions. The worksheet focuses on finding the LCM and HCF, and the accompanying mark scheme breaks down each solution clearly. Suitable for AQA, Edexcel and OCR. Download the questions and answers for free. The HCF uses the shared prime factors; the LCM uses every factor at its highest power.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

The highest common factor of two numbers is the largest number that divides into both. The lowest common multiple is the smallest number that both divide into.

For small numbers, listing works. Write out the factors or multiples of each and pick the largest or smallest shared value. For larger numbers this becomes slow and unreliable.

The efficient method uses prime factorisation. For the HCF, take the primes both numbers share, using the lower power of each. For the LCM, take every prime that appears in either, using the higher power. That approach works whatever the size of the numbers.

Revision notes

Using prime factors for the HCF

Write both numbers as products of primes, then take each shared prime to the lower power.

For \(60 = 2^2 \times 3 \times 5\) and \(72 = 2^3 \times 3^2\): they share \(2\) and \(3\). The lower powers are \(2^2\) and \(3\), so the HCF is \(4 \times 3 = 12\).

Using prime factors for the LCM

Take every prime appearing in either number, using the higher power of each.

For the same numbers: \(2^3\), \(3^2\) and \(5\), giving \(8 \times 9 \times 5 = 360\). Every prime must be included, even those in only one number.

A useful check

For any two numbers, the HCF multiplied by the LCM equals the product of the numbers.

Here \(12 \times 360 = 4320\), and \(60 \times 72 = 4320\). This is a quick way to confirm both answers at once.

Key points

  • The HCF is the largest number dividing into both.
  • The LCM is the smallest number both divide into.
  • For the HCF, take shared primes to the lower power.
  • For the LCM, take all primes to the higher power.
  • HCF × LCM = the product of the two numbers.
  • Prime factorisation is faster than listing for large numbers.

Worked examples

Example 1

Find the HCF of \(24\) and \(36\).

Working

\[24 = 2^3 \times 3 \text{, } 36 = 2^2 \times 3^2\]write both as products of primes
\[2^2 \times 3\]take each shared prime to the lower power
\[= 12\]evaluate the product

Example 2

Find the LCM of \(24\) and \(36\).

Working

\[24 = 2^3 \times 3 \text{, } 36 = 2^2 \times 3^2\]use the same prime factorisations
\[2^3 \times 3^2\]take each prime to the higher power
\[= 72\]evaluate the product

Example 3

Two bells ring every \(12\) and \(18\) minutes. They ring together at noon. When do they next ring together?

Working

\[12 = 2^2 \times 3 \text{, } 18 = 2 \times 3^2\]this asks for the LCM
\[2^2 \times 3^2 = 36\]take the higher power of each prime
\[12{:}36 \text{ pm}\]add 36 minutes to noon

Common mistakes

  • Mixing up the HCF and LCM rules.

    HCF uses shared primes at the lower power; LCM uses all primes at the higher power.

  • Omitting a prime from the LCM.

    Every prime in either number must appear, including those in only one of them.

  • Giving an HCF larger than the smaller number.

    The HCF can never exceed the smaller of the two numbers.

  • Choosing the wrong one in a word problem.

    Events coinciding again means LCM; sharing into equal groups means HCF.

Exam tips

  • Write both prime factorisations out fully before comparing.
  • Use the HCF × LCM check to confirm both answers.
  • In word problems, ask whether things are being grouped (HCF) or repeating (LCM).
  • Sense-check: the HCF is at most the smaller number, the LCM at least the larger.

Key terms

Highest common factor
The largest number that divides exactly into both numbers.
Lowest common multiple
The smallest number that both numbers divide into.
Prime factorisation
Writing a number as a product of primes.
Common
Shared by both numbers.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.