Factors and Multiples
A factor of a number divides into it exactly with no remainder. A multiple of a number is what you get by multiplying it by a whole number - its times-table. The highest common factor (HCF) is the largest number that divides two numbers exactly, and the lowest common multiple (LCM) is the smallest number that appears in both times-tables.
Before you start
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Method
- List the factors of each number by working through pairs that multiply to give it (1 x n, 2 x n/2, and so on) until the pairs start repeating.
- List the first several multiples of each number by counting up in that number's times-table.
- To find the HCF, compare the factor lists and pick the largest number that appears in both.
- To find the LCM, compare the multiple lists and pick the smallest number that appears in both.
- For larger numbers, break each one into prime factors first using a factor tree; the HCF is the product of the primes common to both, and the LCM is the product of every prime that appears, using the highest power seen in either number.
Worked example
Find the HCF and LCM of 18 and 24.
- Factors of 18: 1, 2, 3, 6, 9, 18.
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
- The common factors are 1, 2, 3 and 6, so the HCF is 6.
- Multiples of 18: 18, 36, 54, 72, 90...
- Multiples of 24: 24, 48, 72, 96...
- The smallest number in both lists is 72, so the LCM is 72.
Practice questions
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Exam-style questions
Written in the style of a GCSE exam paper, with a full mark scheme.
Two flashing lights start together. Light A flashes every 12 seconds and light B flashes every 18 seconds. After how many seconds will they next flash together? Show your working.
Write 84 as a product of its prime factors.
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