What are the Multiples of 9?
When 9 is multiplied by any natural number, the resulting product is known as a multiple of 9.
For example: If we multiply 9 by 7, then the product is 63. As a result, we define 63 as a multiple of 9. Another aspect to remember here is that the product of two numbers is a multiple of both the numbers. Hence, 63 is a multiple of 7 as well.
The multiples of 9 can be expressed in the form of 9n, where n is a natural number, such that n = 1, 2, 3, 4, 5, 6, …
Then the multiples of 9 will be: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, …
Also, we can observe that the difference between consecutive multiples of 9, that is, each succeeding and preceding multiple of 9, is 9, and this can be represented as,
18 – 9 = 9,
27 – 18 = 9,
36 – 27 = 9, and so on.
For example, 63, 72 and 81 are all multiples of 9 because we can get these numbers by multiplying 9 with specific natural numbers:
9 × 7 = 63 | 9 is multiplied by 7 to get 63 |
|---|---|
9 × 8 = 72 | 9 is multiplied by 8 to get 72 |
9 × 9 = 81 | 9 is multiplied by 9 to get 81 |
Finding Multiples of 9 by Repeated Addition
Repeated addition means adding a number to itself, multiple times. So we can also use repeated addition to find the multiples of a number.
The n\(^{th}\) Multiple of 9
The n\(^{th}\) multiple of 9 can be obtained by either multiplying 9 by n or by adding 9 n times, where n is a natural number.
n\(^{th}\) multiple of 9 = 9 x n, or,
n\(^{th}\) multiple of 9 = 9 + 9 + … n times
The 9 Times Table
Solved About Multiples of 9 Examples
Example 1: Is 42 a multiple of 9?
Solution:
List multiples of 9.
9, 18, 27, 36, 45, 54, 63, 72, 81, 90, …
42 is not included in the above list.
Also \(\frac{42}{9}\) = 4 R6. When 42 is divided by 9, the remainder is 6. So 9 is not a factor of 42.
Therefore, 42 is not a multiple of 9.
Example 2: What is the 7th multiple of 9?
Solution:
The 7th multiple of 9 can be obtained by the repeated addition of 9, 7 times or multiplying 7 by 9:
9 + 9 + 9 + 9 + 9 + 9 + 9 = 9 x 7 = 63
Hence, the 7th multiple of 9 is 63.
Example 3: Sam, Ricky and John are three friends who decided to buy notebooks in the order of the first three multiples of 9. The cost for each of the books they bought are in the order of the next three multiples of 9. Can you find the total amount they spend?
Solution:
The first three multiples of 9 are 9, 18 and 27. Similarly, the next three multiples of 9 are 36, 45 and 54.
So, Sam bought 9 notebooks and each book costs $36.
Amount spent by Sam = $36 x 9 = $324
Ricky bought 18 notebooks and each book costs $45.
Amount spent by Ricky = $45 x 18 = $810
John bought 27 notebooks and each book costs $54.
Amount spent by John = $54 x 27 = $1458
Total amount spent $324 + $810 + $1458 = $2592
So, the total amount spent by the three friends is $2592.
When a natural number is multiplied by another natural number, the product obtained is a multiple of both the numbers.
If a natural number is multiplied with another natural number then the resultant product is known as a multiple and the multiplied numbers are known as factors of the resultant product.
For example, 9 x 4 = 36
36 is a multiple of both 4 and 9.
4 and 9 are factors of 36.
The smallest multiple of 9 is 9 itself.
There is an infinite set of natural numbers. Hence, we can have an infinite set of multiples for a given number.