What are the Multiples of 21?
The multiples of 21 are the numbers that can be obtained by multiplying 21 by an integer. In other words, if you take 21 and multiply it by 1, 2, 3, and so on, you will get the multiples of 21. The first few multiples of 21 are 21, 42, 63, 84, 105, and so forth. These numbers share the common factor of 21 and are evenly divisible by it.
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The pattern of multiples continues infinitely in both positive and negative directions. For example, in the negative direction, the multiples include -21, -42, -63, and so on. Each of these numbers can be expressed as 21 times an integer.
Multiples are essential in various mathematical concepts, such as arithmetic and algebra. They play a crucial role in finding common denominators, solving equations, and understanding number relationships. In the case of 21, its multiples are useful for calculations involving quantities that are in increments of 21 or for identifying numbers that have 21 as a divisor.
What is a Multiple?
A multiple is a number that can be obtained by multiplying a given number by an integer. In simpler terms, a multiple is a product of a number and an integer. For example, 6 is a multiple of 2 because 2 × 3 = 6.
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Another way to understand multiples is to think of them as the numbers that you skip when counting by a certain number. For instance, if you are counting by 2s, you would skip 1, 3, 5, and so on, because these numbers are not multiples of 2. On the other hand, 2, 4, 6, and so on are all multiples of 2 because you would not skip them when counting by 2s.
Here are some examples of multiples:
- Multiples of 2: 2, 4, 6, 8, 10, ...
- Multiples of 3: 3, 6, 9, 12, 15, ...
- Multiples of 4: 4, 8, 12, 16, 20, ...
- Multiples of 5: 5, 10, 15, 20, 25, ...
As you can see, the multiples of a number continue indefinitely. There is no limit to the number of multiples a number can have.
What is a Common Multiple?
A common multiple of two or more numbers is a number that is a multiple of each of the numbers. In other words, a common multiple is a number that can be obtained by multiplying each of the numbers by an integer. For example, the common multiples of 2 and 3 are 6, 12, 18, and so on.
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The least common multiple (LCM) of two or more numbers is the smallest number that is a common multiple of the numbers. For example, the LCM of 2 and 3 is 6.
Common multiples are often used in arithmetic problems, such as when you need to find a common denominator for two fractions. They can also be used to solve problems involving rates and ratios.
Here are some examples of how to find common multiples:
- Listing multiples: List out the multiples of each number until you find a number that is common to both lists.
- Prime factorization: Factor each number into its prime factorization. Then, multiply the highest power of each prime that appears in any of the factorizations.
- Euclidean algorithm: This is a more advanced method that can be used to find the LCM of two or more numbers.
Properties of a Common Multiple
In arithmetic, a common multiple of two or more integers is an integer that is a multiple of each of the given integers. For example, the common multiples of 4 and 6 are 12, 24, 36, and so on.
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The least common multiple (LCM) of two or more integers is the smallest positive integer that is a common multiple of all the given integers. For example, the LCM of 4 and 6 is 12.
Here are some properties of common multiples and the LCM:
Every common multiple is greater than or equal to the LCM. This is because the LCM is the smallest positive integer that is a common multiple of all the given integers.
The product of the LCM and the highest common factor (HCF) of two integers is equal to the product of the two integers. This is known as the Bezout's identity.
The LCM of two or more integers cannot be less than any of the given integers. This is because the LCM is the smallest positive integer that is a common multiple of all the given integers.
The LCM of two or more integers is always divisible by the HCF of those integers. This is because the HCF is the largest positive integer that is a divisor of all the given integers.
The LCM of two integers where one of the numbers is a prime number is either their product or the larger number itself (this happens when the other number is a multiple of the prime number). For example, the LCM of 4 and 6 is 12, which is the product of the two numbers. The LCM of 4 and 8 is 8, which is the larger number itself.
These properties can be used to solve a variety of problems involving common multiples and the LCM.
List of First 20 Multiples of 21
Here is the list of first 20 multiples of 21:
| Number | Multiple of 21 |
|---|---|
| 1 | 21 |
| 2 | 42 |
| 3 | 63 |
| 4 | 84 |
| 5 | 105 |
| 6 | 126 |
| 7 | 147 |
| 8 | 168 |
| 9 | 189 |
| 10 | 210 |
| 11 | 231 |
| 12 | 252 |
| 13 | 273 |
| 14 | 294 |
| 15 | 315 |
| 16 | 336 |
| 17 | 357 |
| 18 | 378 |
| 19 | 399 |
| 20 | 420 |
Solved Examples on Multiples of 21
Here are some solved examples on multiples of 21:
Example 1: Finding the first 10 multiples of 21
The first 10 multiples of 21 are:
21, 42, 63, 84, 105, 126, 147, 168, 189, 210
Example 2: Finding the next 10 multiples of 21
The next 10 multiples of 21 are:
231, 252, 273, 294, 315, 336, 357, 378, 399, 420
Example 3: Finding the LCM of 21 and 28
The LCM of 21 and 28 is 84.
Example 4: Finding the GCF of 21 and 35
The GCF of 21 and 35 is 7.
