Multiples of a number
Theory
A multiple of a number is the result of multiplying it by a whole number, so multiples are the numbers you say when you skip count. This Year 5 maths page explains how to list multiples, find a particular multiple, find the next multiple after any number, recognise multiples of \(2\), \(5\) and \(10\), and find common multiples and the lowest common multiple.
A multiple of a number is that number multiplied by \(1, 2, 3, 4, \ldots\). The multiples of \(7\) are \(7,\ 14,\ 21,\ 28,\ 35\) and so on. They never stop.
The tenth multiple of \(7\) is \(7 \times 10 = 70\). The first multiple of a number is the number itself.
Factors and multiples are linked: \(7 \times 4 = 28\), so \(28\) is a multiple of \(7\), and \(7\) is a factor of \(28\). Multiples are bigger than (or equal to) the number; factors are smaller than (or equal to) it.
A common multiple of two numbers is a multiple of both. The smallest one is the lowest common multiple (LCM).
Multiples come from multiplication:
| Multiples of | Always end in |
|---|---|
| \(2\) | \(0,\ 2,\ 4,\ 6\) or \(8\) (even) |
| \(5\) | \(0\) or \(5\) |
| \(10\) | \(0\) |
How to find a multiple
- Skip count from the number, or
- Multiply: the \(n\)th multiple of \(a\) is \(a \times n\).
How to test whether a number is a multiple
- Divide it by the number. No remainder means it is a multiple.
- Use endings for \(2\), \(5\) and \(10\).
How to find the lowest common multiple
- List multiples of the larger number.
- Check each one: does the smaller number divide it?
- The first that works is the lowest common multiple.
Skip count by \(9\), or multiply \(9\) by \(1,\ 2,\ 3,\ 4,\ 5,\ 6\).
| \(9 \times 6\) | \(=\) | \(54\) |
\(9,\ 18,\ 27,\ 36,\ 45,\ 54\).
(ii) What is the twelfth multiple of \(5\)?
| \(6 \times 7\) | \(=\) | \(42\) |
| \(5 \times 12\) | \(=\) | \(60\) |
(i) \(42\).
(ii) \(60\).
| \(8 \times 7\) | \(=\) | \(56 \text{ (too small)}\) |
| \(8 \times 8\) | \(=\) | \(64\) |
The next multiple of \(8\) after \(60\) is \(64\).
| \(90 \div 8\) | \(=\) | \(11 \text{ r } 2\) |
| \(8 \times 11\) | \(=\) | \(88\) |
\(88\).
(i) Which are multiples of \(5\)?
(ii) Which is a multiple of \(10\)?
(iii) Which are multiples of \(2\)?
\(135\) ends in \(5\), \(270\) ends in \(0\) and \(412\) ends in \(2\).
(i) \(135\) and \(270\).
(ii) \(270\).
(iii) \(270\) and \(412\).
| \(\text{multiples of } 12\) | \(:\) | \(12,\ 24,\ 36\) |
| \(24 \div 8\) | \(=\) | \(3\) |
\(12\) is not a multiple of \(8\), but \(24\) is. The lowest common multiple is \(24\).
(i) What is the smallest number of each she can buy?
(ii) How many packs of each is that?
| \(\text{multiples of } 10\) | \(:\) | \(10,\ 20,\ 30,\ 40\) |
| \(40 \div 8\) | \(=\) | \(5\) |
| \(40 \div 10\) | \(=\) | \(4\) |
(i) \(40\).
(ii) \(4\) packs of sausages and \(5\) packs of rolls.
| \(9 \times 7\) | \(=\) | \(63\) |
\(9\) is a factor of \(63\). It is \(63\) that is a multiple of \(9\).
Common pitfalls
Frequently asked questions
What is a multiple?
A multiple of a number is that number multiplied by a whole number. The multiples of \(7\) are \(7,\ 14,\ 21,\ 28\) and so on.
What is the difference between a factor and a multiple?
A factor divides into a number; a multiple is made by multiplying it. Since \(9 \times 7 = 63\), \(9\) is a factor of \(63\), and \(63\) is a multiple of \(9\).
How do you find the tenth multiple of a number?
Multiply the number by \(10\). The tenth multiple of \(7\) is \(70\).
How can you tell if a number is a multiple of 5 or 10?
Multiples of \(5\) end in \(0\) or \(5\). Multiples of \(10\) end in \(0\).
What is the lowest common multiple?
It is the smallest number that is a multiple of both numbers. The lowest common multiple of \(8\) and \(12\) is \(24\).
Do multiples ever end?
No. You can always multiply by a bigger whole number, so every number has endless multiples.