How to Find HCF and LCM
The highest common factor (HCF) is the biggest factor two numbers share. The lowest common multiple (LCM) is the smallest multiple they share. Find either by listing or prime factorisation.
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What Are HCF and LCM?
- HCF (Highest Common Factor) is the biggest factor the numbers share
- LCM (Lowest Common Multiple) is the smallest multiple the numbers share
Finding HCF and LCM by Listing
- For HCF, list the factors of each number and choose the largest common one
- For LCM, list the multiples of each number and choose the smallest common one
Finding HCF and LCM by Prime Factorisation
- Use prime factorisation for larger numbers or when listing would be long
- Write each number as a product of prime factors
Using Prime Factors to Find HCF and LCM
- For HCF, take the common primes with the lowest powers
- For LCM, take all primes with the highest powers
Practice Questions
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What is the HCF of 8 and 12?
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The factors of 8 are 1, 2, 4, 8, and the factors of 12 are 1, 2, 3, 4, 6, 12. The highest common factor is 4.
What is the LCM of 4 and 6?
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The multiples of 4 are 4, 8, 12, 16, and the multiples of 6 are 6, 12, 18. The smallest multiple common to both is 12.
What is the LCM of 15 and 20?
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The multiples of 15 are 15, 30, 45, 60, and the multiples of 20 are 20, 40, 60. The smallest common multiple is 60.
What is the HCF of 45 and 75?
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The factors of 45 are 1, 3, 5, 9, 15, 45, and the factors of 75 are 1, 3, 5, 15, 25, 75. The largest common factor is 15.
Using prime factorisation, what is the LCM of 28 and 42?
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For 28, the prime factorisation is 2 × 2 × 7. For 42, it is 2 × 3 × 7. The LCM is .
What is the LCM of 72 and 108?
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The prime factorisation of 72 is 2³ × 3², and the prime factorisation of 108 is 2² × 3³. Taking the highest powers of each prime gives .
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Interactive Activity
Visualise HCF and LCM using prime factorisation
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The HCF, or Highest Common Factor, is the largest factor that two or more numbers share. The LCM, or Lowest Common Multiple, is the smallest multiple they share. One looks for the biggest factor the numbers have in common, the other for the smallest multiple they have in common.
First write each number as a product of its prime factors. For the HCF, multiply the prime factors that both numbers share, taking each one at its lowest power. For the LCM, multiply every prime factor that appears across the numbers, taking each one at its highest power.
Use listing when the numbers are small, because writing out their factors and multiples stays quick and manageable. For larger numbers, listing every factor and multiple becomes less convenient, so prime factorisation is the more efficient and reliable method to reach for.
The HCF of 12 and 18 is 6, the largest factor they share. The LCM of 12 and 18 is 36, the smallest multiple they share. You can find both by listing the factors and multiples of each number and spotting the values they have in common.
Yes. The prime factorisation method works for any pair or group of whole numbers, whether they are small or large. It is especially useful for larger numbers, where listing would take a long time. The method stays efficient and reliable every time you use it.