What Is A Multiple? Multiples are fundamental to understanding math concepts. At WHAT.EDU.VN, we provide a clear explanation of multiples, offering a solution to grasp this essential topic effortlessly. Delve into properties of multiples, factor relationships, and practical examples.
1. What Are Multiples?
In mathematics, the definition of a multiple refers to the product you get when you multiply a whole number by another whole number. Think of it as simply using your multiplication tables! For instance, let’s find the first five multiples of 6 (excluding zero). They are 6, 12, 18, 24, and 30. Notice that these are the numbers you find in the “table of 6.”
Therefore, the key takeaway is:
Multiple of a number = Number × Any integer (not a fraction).
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2. Listing Multiples
You can generate multiples of any number by multiplying it by integers, both positive and negative. It’s important to note that a number has an infinite number of multiples. Below is a table illustrating the first ten non-zero multiples for several numbers:
| Number | First Ten Multiples (Non-Zero) |
|---|---|
| 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 |
| 10 | 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 |
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3. Understanding Properties of Multiples
These properties offer insights into the nature of multiples:
3.1. Every Number Is A Multiple Of Itself
For instance, 7 is the first non-zero multiple of 7 because 7 × 1 = 7. This illustrates a fundamental property of multiples.
3.2. Multiples Are Infinite
Since numbers go on infinitely, any number will also have an infinite number of multiples. When listing multiples of 3, you start with 3, 6, 9, 12, 15, 18, and so on. You can’t list all of them because they never end.
3.3. Multiples Are Greater Than Or Equal To The Original Number (Except Zero)
Take the multiples of 5: 5, 10, 15, 20, 25, 30, and so on. The first multiple is 5 × 1 = 5, which equals the original number. All subsequent multiples (10, 15, 20…) are greater than 5.
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4. The Connection Between Factors and Multiples
Factors and multiples are intertwined concepts.
- A factor divides another number evenly, leaving no remainder.
- A multiple is the product of one number multiplied by another.
For example, in the equation 3 × 4 = 12:
- 3 and 4 are factors of 12.
- 12 is a multiple of both 3 and 4.
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5. Exploring Common Multiples
A common multiple is a number that appears as a multiple of two or more different numbers.
For example:
- Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30…
- Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36…
In these lists, 12 and 24 are common multiples of both 3 and 4.
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6. Practical Multiple Examples
Let’s solidify your understanding with a few examples:
6.1. Example 1
Four friends, Ria, Joe, Sam, and Tom, decided to collect flowers in the order of the first four multiples of 7 (excluding zero). How many flowers did each friend collect?
Solution:
The first four multiples of 7 are:
- (7 × 1) = 7
- (7 × 2) = 14
- (7 × 3) = 21
- (7 × 4) = 28
Therefore:
- Ria collected 7 flowers.
- Joe collected 14 flowers.
- Sam collected 21 flowers.
- Tom collected 28 flowers.
6.2. Example 2
Ann loves watering plants. Her teacher asked her to water the pots marked with multiples of 9. She missed some pots. Can you identify the missing numbers in this list: 9, 18, __, 36, __, 54, 63, 72, __?
Solution:
Let’s use the multiplication table of 9:
- 9 × 1 = 9
- 9 × 2 = 18
- 9 × 3 = 27
- 9 × 4 = 36
- 9 × 5 = 45
- 9 × 6 = 54
- 9 × 7 = 63
- 9 × 8 = 72
- 9 × 9 = 81
Therefore, the missing pots are marked 27, 45, and 81.
6.3. Example 3
List the first five multiples of 12.
Solution:
Multiply 12 by the first five natural numbers:
- 12 × 1 = 12
- 12 × 2 = 24
- 12 × 3 = 36
- 12 × 4 = 48
- 12 × 5 = 60
Therefore, the first five multiples of 12 are 12, 24, 36, 48, and 60.
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7. Frequently Asked Questions (FAQs) About Multiples
Let’s address some common questions about multiples:
| Question | Answer |
|---|---|
| 7.1. What are multiples in math? | Multiples in math are numbers obtained by multiplying a whole number by another whole number. For example, in 7 × 5 = 35, 35 is a multiple of 7 and 5. |
| 7.2. How are factors and multiples related? | A factor divides another number completely without a remainder. A multiple is the result of multiplying one number by another. In 6 × 8 = 48, 6 and 8 are factors of 48, while 48 is a multiple of 6 and 8. |
| 7.3. What are common multiples? | Common multiples are shared multiples of two or more numbers. For example, multiples of 3 include 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, and multiples of 5 include 5, 10, 15, 20, 25, 30. 15 and 30 are common multiples of 3 and 5. There are infinitely many common multiples for any set of numbers. You can find common multiples by finding multiples of the Least Common Multiple (LCM) of the numbers. |
| 7.4. How to find multiples of a number? | You find multiples by multiplying a number by another number. For example, if 8 × 5 = 40, 40 is a multiple of 8 and 5. Other multiples of 8 are: 8 (8 × 1 = 8), 16 (8 × 2 = 16), 24 (8 × 3 = 24), and so on. |
| 7.5. How many multiples does a number have? | A number has an infinite number of multiples. There is no limit. |
| 7.6. What are the multiples of 5? | The multiples of 5 are: 5 (5 × 1 = 5), 10 (5 × 2 = 10), 15 (5 × 3 = 15), and so on. You can list them as: 5, 10, 15, 20, 25, and so on. |
| 7.7. What are the multiples of 4? | The multiples of 4 are 4, 8, 12, 16, 20, and so on. These are calculated by multiplication: 4 (4 × 1 = 4), 8 (4 × 2 = 8), 12 (4 × 3 = 12), and so on. |
| 7.8. What are the multiples of 2? | The multiples of 2 are 2, 4, 6, 8, 10, 12, and so on. This is also the list of all even numbers, as all even numbers are multiples of 2. |
| 7.9. How do you explain multiples simply? | Multiples are the products you get when multiplying one whole number by another whole number. For example, 2 × 3 = 6, so 6 is a multiple of 3 and 2. Other multiples of 2 include 2, 4, 6, 8, 10, 12, and so on. |
| 7.10. What are the multiples of 3? | The multiples of 3 are 3, 6, 9, 12, 15, and so on. They are calculated by multiplying 3 with counting numbers: 3 (3 × 1 = 3), 6 (3 × 2 = 6), 9 (3 × 3 = 9), and so on. |
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