Numbers and sequences
Common multiples & factors
A multiple of a number is any number in its times-table. A factor divides exactly into a number. When two numbers share a multiple or a factor, we call it a COMMON multiple or a COMMON factor.
Multiples and common multiples
The multiples of a number are the numbers in its times-table. The multiples of 3 are 3, 6, 9, 12, 15, ... A common multiple is a number that appears in the times-tables of two (or more) numbers. For example, 24 is a common multiple of 2, 3, 4, 6, 8, 12 and 24 because they all divide into it.
Factors and common factors
A factor of a number divides into it exactly, with no remainder. The factors of 18 are 1, 2, 3, 6, 9 and 18. A common factor is a number that is a factor of two numbers. For example, 6 is a common factor of 18 and 24.
Finding a common multiple
To find a common multiple, list the multiples of each number and look for one that appears in both lists. Multiples of 3: 3, 6, 9, 12, 15, 18. Multiples of 5: 5, 10, 15. The number 15 is in both lists, so it is a common multiple of 3 and 5.
Finding a common factor
To find a common factor, list the factors of each number and look for the ones that appear in both lists. 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.
Worked examples
Find a common multiple (Worked example 3)
- List multiples of 3: 3, 6, 9, 12, 15, 18, and multiples of 5: 5, 10, 15.
- Find the number that is in both lists and less than 20.
Answer: 15 is a common multiple of 3 and 5 that is less than 20.
Find common multiples under a limit
- List multiples of 8 under 50: 8, 16, 24, 32, 40, 48.
- Keep only the ones that are also multiples of 3: check 24 and 48.
Answer: The common multiples of 3 and 8 under 50 are 24 and 48.
Find common factors
- List factors of 18: 1, 2, 3, 6, 9, 18, and factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
- Ring the numbers that appear in both lists.
Answer: The common factors of 18 and 24 are 1, 2, 3 and 6.
Common multiples in real life
- Olivia plays every 4 days (days 4, 8, 12) and Amir every 3 days (days 3, 6, 9, 12).
- A day they both play is a common multiple of 3 and 4; find one within 14 days.
Answer: They next play together on day 12, so once in the fortnight.
Where you see this in real life
- Working out when two friends who train on different day-cycles will train together again.
- Sharing a bag of sweets equally between friends so nobody is left out (using factors).
- Setting a table so that rows of chairs and rows of plates line up (using common multiples).
Key words
- Multiple
- A number in another number's times-table, such as 12 is a multiple of 3.
- Factor
- A number that divides exactly into another number, such as 4 is a factor of 12.
- Common multiple
- A number that is a multiple of two or more numbers.
- Common factor
- A number that is a factor of two or more numbers.