In Mathematics, multiples are the numbers we get when we multiply a number by other natural numbers, and they play a significant role in our daily life. Understanding the concept of multiples eases our day-to-day life with simple calculations and effective results.
Suppose you and your friend are going to your favorite coffee shop. You went there for breakfast every third day, and your friend has his breakfast every fifth day. How many days would it take for you to meet again for breakfast at the same restaurant?
The answer is fifteen. But how did we come up with this answer? This is the magic of multiples. And in this article, we will be learning another group of multiples: the multiples of 9.
Are you ready to delve into the world of the multiples of 9? Be prepared for we will start another exciting mathematical journey!
Multiples of 9 are the numbers 9, 18, 27, 36, 45, 54, 63, 72, 81, 90…
Notice that the set of numbers generates a sequence wherein the difference of two consecutive numbers in the sequence is 9. Also, the numbers will result in a whole number when divided by 9.
In other words, a multiple of 9 is the product of 9 and any integer. When we multiply 9 with any positive integer, the product is a positive multiple of 9, and when we multiply 9 with any negative integer, the product will be a negative multiple of 9.
Remember that fractions cannot be used in finding multiples.
For example, 36 is a multiple of 9, as it is the product of 9 and 4. Other examples of multiples of 9 are 9, 27, 54, 99, 891, and so on.
How to Find the Multiples of 9?
We can determine the multiples of 9 by repeated addition. We just need to add 9 as many times as necessary. Let’s take a look at the first five multiples of 9 using repeated addition or skip counting.
Alternatively, we can get the nth multiple of 9, by multiplying 9 by n.
Say, for example, we are asked to get the 4th multiple of 9.
For us to determine the fourth multiple of 9, we must multiply 9 by 4, that is, $$9\;\times\;4\;=\;36$$ .
Isn’t that easy? Let us try another one and find the 7th multiple of 9.
By multiplication, the seventh multiple of 9 is given by $$9\;\times\;7\;=\;63$$ . Thus, the seventh multiple of 9 is 63.
The table below shows the comparison of the multiplication and repeated addition in determining the multiples of 9.
| nth multiple of 9 | by Multiplication | by Repeated Addition | Multiples of 9 |
|---|---|---|---|
| 1st multiple | 9 × 1 | 9 | 9 |
| 2nd multiple | 9 × 2 | 9 + 9 | 18 |
| 3rd multiple | 9 × 3 | 9 + 9 + 9 | 27 |
| 4th multiple | 9 × 4 | 9 + 9 + 9 + 9 | 36 |
| 5th multiple | 9 × 5 | 9 + 9 + 9 + 9 + 9 | 45 |
Now, you’ll be able to find any number that is a multiple of 9!
List of the First 30 Multiples of 9
There are infinitely many multiples of 9, for there are also infinitely many integers. Below are the first 30 multiples of 9 by which are obtained by multiplying 9 by each of the natural numbers from 1 to 30.
| Product of 9 and a positive counting number | Multiples 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 |
| 9 × 10 | 90 |
| 9 × 11 | 99 |
| 9 × 12 | 108 |
| 9 × 13 | 117 |
| 9 × 14 | 126 |
| 9 × 15 | 135 |
| 9 × 16 | 144 |
| 9 × 17 | 153 |
| 9 × 18 | 162 |
| 9 × 19 | 171 |
| 9 × 20 | 180 |
| 9 × 21 | 189 |
| 9 × 22 | 198 |
| 9 × 23 | 207 |
| 9 × 24 | 216 |
| 9 × 25 | 225 |
| 9 × 26 | 234 |
| 9 × 27 | 243 |
| 9 × 28 | 252 |
| 9 × 29 | 261 |
| 9 × 30 | 270 |
Have you noticed that the multiples of the number 9 in the table above are also the results in the multiplication table of 9? It is because both are the same.
Did you know that…
When you add up all the digits of a multiple of 9, the sum is also a multiple of 9?
Take a look at the first ten multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, and 90. When you add the digits of each number, the sum is always 9.
Cool, right? Let us check other multiples of 9 with higher values: 135, 1566, and 42957.
The sum of the digits of 135 is $$1\;+\;3\;+\;5\;=\;9$$ . The sum of the digits is 9, which is a multiple of itself. The sum of the digits of 1566 is $$1\;+\;5\;+\;6\;+\;6\;=\;18$$ . 18 is a multiple of 9, and the sum of its digits is also 9.
The sum of the digits of 42957 is $$4\;+\;2\;+\;9\;+\;5\;+\;7\;=27$$ . 27 is a multiple of 9 and, the sum of its digits is also 9.
What an amazing discovery! Now we can say that if the sum of all the digits of a number is a multiple of 9 then, that number is a multiple of 9 and is, therefore, divisible by 9.
Try it! Look for more multiples of 9 and see that this is always true.
Problems Involving Multiples of 9
Let us now try what we have learned about the multiples of 9 in solving real-life problems.
Problem #1
John landed a job in an appliance center where he earns $9 a day. If he had already worked there for six consecutive days, how much did he already earn?
We know that John earns $9 per day. To determine the total earnings of John for six days, we need to find the 6th multiple of his daily salary.
So, $$9\;\times\;6\;=\;54$$ .
Therefore, John already earned $54.
Problem #2
Mary and eight of her friends went to a farm where they could pick some strawberries. They decided to combine all the strawberries they had gathered and then split the total equally.
If they gather a total of 234 strawberries, is it possible for Mary and her friends to have equal shares of the strawberries they collected?
There is a total of 9 persons, Mary and her eight friends, who will share the 234 strawberries they collected. To determine if they can split the total number of strawberries equally, we need to know if 234 is a multiple of 9.
We have learned that the sum of the digits of a multiple of 9 is also a multiple of 9. Let’s add the digits of 234, $$2\;+\;3\;+\;4\;=\;9$$. The sum of the digits is 9, and 9 is a multiple of itself.
Therefore, Mary and her friends can have equal shares of the strawberries they collected. Going further, dividing 234 by 9, each of them will have 26 strawberries!
How simple is that? Learning about multiples is really useful to most real-life situations!
Take a Quiz!
Let us put your understanding of the concept of multiples of 9 to test by answering the following problems.
