What Are Multiples?
A multiple of a number is the product of multiplying that number by another number.
The multiples of 4 are 4, 8, 12, 16, 20,…..
5 x 3 = 15. Here, 15 is a multiple of both 3 and 5.
Properties of Multiples
- A number is always a multiple of itself.
- The smallest multiple of a number is the number itself.
- There is no largest multiple of a number.
- A number has infinite multiples.
- Every number is a multiple of 1.
How to Identify Multiples?
To find multiples of any number, start with that number and keep adding it to itself. For example, to get multiples of 5:
- Start with 5.
- Add 5 repeatedly: 5, 10, 15, 20, 25, and so on.
This pattern of addition (or repeated multiplication) is why we sometimes call them “skip-counting.”
Differences Between Factors and Multiples
While multiples are products formed by multiplying, factors are numbers that divide another number evenly without leaving a remainder.
For instance, for the number 12:
- Multiples of 12: 12, 24, 36, 48, etc.
- Factors of 12: 1, 2, 3, 4, 6, 12.
Understanding these differences helps with multiplication, division, and breaking down numbers into smaller parts.
Real-World Examples of Multiples
Multiples are everywhere in real life:
- Counting Events: If a bus arrives every 10 minutes, the multiples of 10 tell us when it will arrive: 10, 20, 30 minutes, etc.
- Grocery Shopping: If packs of items come in multiples of 4, you get 4, 8, 12 items by buying one, two, or three packs.
- Time Management: Planning events in blocks of 15 minutes (multiples of 15) helps organize schedules.
Common Multiples
Common multiples are numbers that are multiples of two or more numbers.
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …
The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, …
The common multiples of 4 and 6 are 12, 24, 36,…
Least Common Multiple (LCM)
The least common multiple (LCM) is helpful when you need to find a shared multiple between numbers, often used for adding fractions or finding common patterns in schedules.
Example:
- Multiples of 6: 6, 12, 18, 24, 30…
- Multiples of 8: 8, 16, 24, 32…
The LCM of 6 and 8 is 24 because it’s the smallest multiple they have in common.
Multiples in Fraction Operations
Multiples are particularly important when working with fractions. To add or subtract fractions, you often find the LCM of the denominators to create a common base.
For example: To add 1/3 and 1/4, you find the LCM of 3 and 4, which is 12. You can then express both fractions with a denominator of 12:
- 1/3 = 4/12
- 1/4 = 3/12
This makes it easy to add: 4/12 + 3/12 = 7/12.
Practice Problems
Test your skills with these practice questions:
- List the first ten multiples of 8.
- What is the LCM of 9 and 12?
- Are 36 and 48 common multiples of 6 and 12? Why or why not?
- If a bell rings every 4 minutes and another rings every 6 minutes, when will they ring together within the first hour?
Practice Quiz on Multiples
FAQs on Multiples
- Yes, multiples of any non-zero number are infinite because you can keep multiplying by larger integers.
- The smallest positive multiple of any non-zero number is the number itself.
- Example: The smallest multiple of 7 is 7.
- Yes, multiples can be negative if multiplied by negative integers.
- Example: Multiples of 3 include −3,−6,−9,…
When we talk about the smallest multiple of a number being the number itself, we are specifically referring to positive multiples.
However, mathematically, multiples of a number can extend in both directions (positive and negative). That’s why it’s possible to have negative multiples.
In the context of positive integers, the smallest multiple is indeed the number itself. But if you include negatives, there’s no smallest multiple because you can keep multiplying by larger negative integers (e.g., −1,−2,−3,…).
In summary:
The “smallest multiple” refers to the smallest positive multiple unless explicitly stated to include negatives.
Negative multiples exist but are usually not considered when talking about the smallest multiple in basic arithmetic.
- No, the absolute value of a non-zero number’s multiples is always greater than or equal to the original number.
- Yes, you can have multiples of decimal numbers by multiplying them by integers.
- Example: Multiples of 0.5 are 0.5,1.0,1.5,…
- No, different numbers will have different sets of multiples, though they may have some common multiples.