Highest common factor and lowest common multiple
Objective. Find HCF and LCM and apply them to scheduling problems.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
The highest common factor (HCF) of two numbers is the largest number that divides into both exactly, while the lowest common multiple (LCM) is the smallest number that both divide into exactly. Using prime factors makes both easy to find: write each number as a product of primes, then the HCF takes the lowest power of each shared prime, and the LCM takes the highest power of every prime involved. For example, with 12=2²×3 and 18=2×3², the HCF is 2×3=6 and the LCM is 2²×3²=36. HCF is useful for sharing things into equal groups, while LCM suits timing problems like buses leaving together.
Worked example
Find the HCF and LCM of 24 and 36.
- 1Write prime factorisations: 24=2³×3, 36=2²×3².
- 2For HCF, take the lowest power of each prime: 2²×3=12.
- 3For LCM, take the highest power of each prime: 2³×3²=72.
- 4Check: 12 divides both 24 and 36; 72 is a multiple of both.
Answer: HCF = 12, LCM = 72
Concept mastery
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Practice
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Start practisingCommon mistake. Pupils often swap HCF and LCM, picking the larger result when asked for HCF. Always check the HCF divides neatly into both original numbers, and the LCM is a multiple of both.
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