What are multiples and factors?
A multiple is a number that can be divided by another number a certain number of times without a remainder.
A factor is one of two or more numbers that divides a given number without a remainder.
Multiples and factors are best explained by using a number sentence such as the following:
This number sentence tells us that 20 is a multiple of 5 and is also a multiple of 4.
It also tells us that 5 and 4 are factors of 20.
Multiples and factors in KS1 and KS2
In Year 2, children are expected to recognise multiples of:
- 2 (these are 2, 4, 6, 8, 10, 12 etc.)
- 5 (these are 5, 10, 15, 20, 25 etc.)
- and 10 (these are 10, 20, 30, 40, 50 etc.).
The foundation for this knowledge is started in Year 1 where children practise counting in 2s, 5s and 10s.
In Year 3, children need to be able to recognise multiples of 2, 5 and 10 up to 1000.
They would be taught the rule that any number that has a last digit that is even (24, 68, 102. 888 etc) is a multiple of 2. They would also learn that any number ending in a 5 or 0 (25, 300, 605, 990 etc.) is a multiple of 5. And finally, any number that ends in a 0 (20, 800, 450 etc.) is a multiple of 10.
By Year 4, when children have learnt all their times tables they should be able to recognise multiples of any number up to 10.
For example, they may be given a number such as 24 and asked if it is a multiple of the following numbers: 10, 2, 5, 8, 6. Their knowledge of times tables should help them to work out that 24 is a multiple of 2, 8 and 6.
Children also learn about factors in Year 4.
They may be asked to identify pairs of factors of two-digit numbers.
- For example: they may be given the number 60 and asked to pick out two pairs of factors from the following numbers: 7, 3, 9, 8, 15, 4, 20. (The answer to this is 3 x 20 and 4 x 15.)
- They may also need to find common multiples. For example: they may be given the numbers 6 and 9 and asked to find three multiples that have both these numbers as factors. (The three numbers could be: 54, 18 and 36.)
In Year 5, children will continue to carry out tasks involving factors and multiples and this will extend to learning about prime numbers and square numbers.
Example investigations for Year 6 children could be:
Two factors of 90 are added together to make another factor of 90. What are the two factors?
What am I? I am a multiple of 4. I am between 25 and 50. I am also a multiple of 10.
Two square numbers are added together to make a multiple of 9. What could the two square numbers be?
By Year 6 children need to be really confident not only on knowing what factors and multiples are, but also on quickly working out the factors and multiples of various numbers. This means they can take quickly to these sorts of puzzles. They might able be asked to find the lowest common multiple or highest common factor of two numbers or a group of numbers.
The least common multiple of two or more non-zero whole numbers is actually the smallest whole number that is divisible by each of the numbers.
There are two widely used methods.
Simply list the multiples of each number (multiply by 2, 3, 4, etc.) then look for the smallest number that appears in each list.
Find the least common multiple for 5, 6, and 15.
- First we list the multiples of each number.
Multiples of 5 are 10, 15, 20, 25, 30, 35, 40,…
Multiples of 6 are 12, 18, 24, 30, 36, 42, 48,…
Multiples of 15 are 30, 45, 60, 75, 90,….
- Now, when you look at the list of multiples, you can see that 30 is the smallest number that appears in each list.
- Therefore, the least common multiple of…
… 5, 6 and 15 is 30.
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To use this method factor each of the numbers into primes. Then for each different prime number in all of the factorizations, do the following…
- Count the number of times each prime number appears in each of the factorizations.
- For each prime number, take the largest of these counts.
- Write down that prime number as many times as you counted for it in step 2.
- The least common multiple is the product of all the prime numbers written down.
Find the least common multiple of 5, 6 and 15.
Factor into primes
Prime factorization of 5 is 5
Prime factorization of 6 is 2 x 3
Prime factorization of 15 is 3 x 5
Notice that the different primes are 2, 3 and 5.
- Now, we do Step #1 – Count the number of times each prime number appears in each of the factorizations…
The count of primes in 5 is one 5
The count of primes in 6 is one 2 and one 3
The count of primes in 15 is one 3 and one 5
- Step #2 – For each prime number, take the largest of these counts. So we have…
The largest count of 2s is one
The largest count of 3s is one
The largest count of 5s is one
- Step #3 – Since we now know the count of each prime number, you simply – write down that prime number as many times as you counted for it in step 2.Here they are…
2, 3, 5
- Step #4 – The least common multiple is the product of all the prime numbers written down.
2 x 3 x 5 = 30// Therefore, the least common multiple of 5, 6 and 15 is 30.
So there you have it.
A quick and easy method for finding least common multiples.
Prime Factorization Table
Lowest common multiple (LCM)