Multiples of a Number and Common Multiples
Multiples of a number are that number multiplied by 1, 2, 3 and every other whole number. The multiples of 6 are 6, 12, 18, 24, 30and so on. Common multiples are the numbers that appear in the multiples of two or more numbers; the smallest one is the least common multiple (LCM).
What Is a Multiple of a Number?
A multiple of n is n × k for a whole number k. The multiples of 7 are 7, 14, 21, 28, 35, 42, and the list never ends. A number is a multiple of n when dividing it by n leaves no remainder.
Two facts follow from the definition:
- Every number is a multiple of itself (n × 1) and of 1.
- The multiples of a number grow by that number each time: 7, 14, 21 add 7 at every step.
Multiples vs Factors: What Is the Difference?
A multiple is the result of multiplying a number; a factor divides into it. 24 is a multiple of 6, and 6 is a factor of 24.
| Multiples of 12 | Factors of 12 | |
|---|---|---|
| Examples | 12, 24, 36, 48, 60 … | 1, 2, 3, 4, 6, 12 |
| How many | Infinitely many | A fixed number (6 for 12) |
| Size | 12 or larger | 12 or smaller |
| Leads to | The LCM (least common multiple) | The HCF / GCF (highest common factor) |
The two ideas meet in one formula, explained on LCM vs HCF (GCF): LCM × HCF = a × b for two numbers.
Multiples of 1 to 20
The first 10 multiples of every number from 1 to 20:
| Number | First 10 multiples |
|---|---|
| 1 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 |
| 2 | 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 |
| 3 | 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 |
| 4 | 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 |
| 5 | 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 |
| 6 | 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 |
| 7 | 7, 14, 21, 28, 35, 42, 49, 56, 63, 70 |
| 8 | 8, 16, 24, 32, 40, 48, 56, 64, 72, 80 |
| 9 | 9, 18, 27, 36, 45, 54, 63, 72, 81, 90 |
| 10 | 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 |
| 11 | 11, 22, 33, 44, 55, 66, 77, 88, 99, 110 |
| 12 | 12, 24, 36, 48, 60, 72, 84, 96, 108, 120 |
| 13 | 13, 26, 39, 52, 65, 78, 91, 104, 117, 130 |
| 14 | 14, 28, 42, 56, 70, 84, 98, 112, 126, 140 |
| 15 | 15, 30, 45, 60, 75, 90, 105, 120, 135, 150 |
| 16 | 16, 32, 48, 64, 80, 96, 112, 128, 144, 160 |
| 17 | 17, 34, 51, 68, 85, 102, 119, 136, 153, 170 |
| 18 | 18, 36, 54, 72, 90, 108, 126, 144, 162, 180 |
| 19 | 19, 38, 57, 76, 95, 114, 133, 152, 171, 190 |
| 20 | 20, 40, 60, 80, 100, 120, 140, 160, 180, 200 |
How Do You Find Common Multiples of Two Numbers?
Find the LCM of the two numbers, then multiply it by 1, 2, 3 and so on. Every common multiple of two numbers is a multiple of their LCM, so this lists them all in order without writing long rows of multiples.
For 6 and 14, by listing:
- Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84
- Multiples of 14: 14, 28, 42, 56, 70, 84
- In both lists: 42, 84
By the shortcut: LCM(6, 14) = 42, so the common multiples are 42, 84, 126 and so on. The same works for 15 and 20: LCM = 60, common multiples 60, 120, 180. The methods for the LCM itself are on How to Find the LCM.
What Are the First Three Common Multiples of Two Numbers?
The first three common multiples are LCM × 1, LCM × 2 and LCM × 3. The table lists them for pairs that come up in homework.
| Numbers | LCM | First three common multiples |
|---|---|---|
| 6 and 14 | 42 | 42, 84, 126 |
| 15 and 20 | 60 | 60, 120, 180 |
| 12 and 15 | 60 | 60, 120, 180 |
| 21 and 28 | 84 | 84, 168, 252 |
| 4 and 6 | 12 | 12, 24, 36 |
| 8 and 12 | 24 | 24, 48, 72 |
| 9 and 12 | 36 | 36, 72, 108 |
| 10 and 15 | 30 | 30, 60, 90 |
Common Multiples of Three Numbers
Three numbers work the same way: find the LCM of all three, then multiply it by 1, 2, 3.
| Numbers | LCM | First three common multiples |
|---|---|---|
| 6, 10, 15 | 30 | 30, 60, 90 |
| 14, 21, 28 | 84 | 84, 168, 252 |
| 2, 5, 6 | 30 | 30, 60, 90 |
| 5, 8, 12 | 120 | 120, 240, 360 |
Multiples of a Number Less Than a Limit
List the multiples in order and stop before the limit. The multiples of 2 less than 15 are 2, 4, 6, 8, 10, 12, 14, which makes 7 in all. Any three of these answer "give three multiples of 2 that are less than 15", for example 2, 4 and 6.
To count them without listing, divide the largest allowed number by n and drop the remainder: 14 ÷ 2 = 7.
How to Check Whether a Number Is a Multiple
Divide by the number; a remainder of 0 means it is a multiple. 75 ÷ 15 = 5 with no remainder, so 75 is a multiple of 15. 80 ÷ 15 leaves 5, so 80 is not.
Quick tests for small numbers: a multiple of 2 ends in 0, 2, 4, 6 or 8; of 5 ends in 0 or 5; of 3 has digits that add up to a multiple of 3; of 9 has digits that add up to a multiple of 9.
When Is the Question Really About One Number?
"LCM of 12" on its own asks for the least common multiple of one number, which is the number itself; the reason is onLCM Edge Cases: One Number, Zero, and Negative Numbers. Most of the time the question means the multiples of 12, listed above, or the common multiples of 12 and a second number.
Common multiples lead straight to the least common multiple, defined on What is the LCM.
Least Multiple Calculator: find the LCM of 2 to 15 numbers, with the multiples listed as one of six methods.
Frequently asked questions
What are the multiples of 12?
The multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108, 120 and so on: 12 × 1, 12 × 2, 12 × 3, without end.
What are the common multiples of 12 and 18?
The common multiples of 12 and 18 are 36, 72, 108 and so on. The smallest, 36, is the LCM of 12 and 18, and every other common multiple is a multiple of 36.
How many multiples does a number have?
Every nonzero whole number has infinitely many multiples, because the list n × 1, n × 2, n × 3 never ends. A number has only a limited number of factors.
Is a number a multiple of itself?
Yes. Every number is a multiple of itself, because n × 1 = n. It is also the smallest positive multiple of that number.
What is the first common multiple of 6 and 14?
The first common multiple of 6 and 14 is 42, the least common multiple. The next two are 84, 126.
Can two numbers have no common multiple?
No. Any two nonzero whole numbers have common multiples, because their product a × b is always one. The LCM is the smallest of them and is never larger than a × b.
Sources
- Wikipedia: Least common multiple: definition, notation, LCM × GCD = a × b
- Wolfram MathWorld: Least Common Multiple: formal properties
- Common Core 6.NS.B.4: GCF and LCM in grade 6