Maths
What are HCF and LCM?
Quick answer
The HCF is the largest number that divides exactly into two numbers. The LCM is the smallest number that both divide into. For 12 and 18, the HCF is 6 and the LCM is 36.
In one sentence
The HCF is the biggest number that goes into both numbers, and the LCM is the smallest number that both go into.
Explained
Two ideas make this work. A factor of a number divides into it exactly (the factors of 12 are 1, 2, 3, 4, 6 and 12). A multiple is in its times table (multiples of 4 are 4, 8, 12, 16 …). See factors and multiples.
Highest common factor (HCF). List the factors of each number and pick the biggest one that appears in both lists.
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- Common factors: 1, 2, 3, 6. The HCF is 6.
Lowest common multiple (LCM). List the multiples of each number until one appears in both lists.
- Multiples of 4: 4, 8, 12, 16
- Multiples of 6: 6, 12, 18
- The LCM of 4 and 6 is 12.
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There is a shortcut using prime factorisation. Write each number as a product of primes. 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3. For the HCF, multiply the primes they share: 2 × 3 = 6. For the LCM, take each prime the greatest number of times it appears in either number: 2 × 2 × 3 × 3 = 36.
There is also a check: for any two numbers, HCF × LCM = the two numbers multiplied. For 12 and 18, 6 × 36 = 216 and 12 × 18 = 216.
Why does it matter? The HCF helps you simplify fractions: 12/18 = 2/3, after dividing by 6. The LCM gives you a common denominator when adding fractions, and solves timing puzzles, such as when two buses meet again.
Examples
| Numbers | HCF | LCM |
|---|---|---|
| 4 and 6 | 2 | 12 |
| 12 and 18 | 6 | 36 |
| 8 and 12 | 4 | 24 |
| 6 and 8 | 2 | 24 |
| 7 and 9 | 1 | 63 |
Common mistakes
- Mixing up the two. HCF is a factor, so it is never bigger than the smaller number. LCM is a multiple, so it is never smaller than the bigger number.
- Stopping the list too early. Check you have listed all the factors.
- Choosing a common multiple that is not the lowest. 24 works for 4 and 6, but the lowest is 12.
Quick quiz
Highest common factor and lowest common multiple
Check what you know with 4 quick questions.
Question 1 of 4
What is the HCF of 12 and 18?
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Related topics
Factors and multiples
Factors are numbers that divide exactly into a number (the factors of 12 are 1, 2, 3, 4, 6 and 12). Multiples are the numbers in its times table (12, 24, 36…).
Prime numbers
A prime number is a whole number greater than 1 that has exactly two factors, 1 and itself. The first primes are 2, 3, 5, 7, 11 and 13.
Prime factorisation
Prime factorisation means writing a number as a product of prime numbers. For example, 60 = 2 × 2 × 3 × 5, which can be written as 2² × 3 × 5.
Powers and indices
A power (also called an index or exponent) is a small raised number showing how many times to multiply a number by itself. So 2⁴ means 2 × 2 × 2 × 2 = 16.
Scientific notation
Scientific notation writes very large or very small numbers as a number from 1 to 10 times a power of 10. For example, 3,500,000 is 3.5 × 10⁶. In the UK it is called standard form.
Square roots
The square root of a number is the value that, multiplied by itself, gives that number. The square root of 25 is 5, because 5 × 5 = 25. The sign is √.