Multiples of 9
Did you know that the sum of all the digits of the multiples of 9 add up to 9. For example, 18 is a multiple of 9 and 1 + 8 = 9. Similarly, 198 is a multiple of 9 and 1 + 9 + 8 = 18 and 1 + 8 = 9. Isn't this interesting? In this mini-lesson, we will calculate the multiples of 9 and we will learn some interesting facts about these multiples with solved examples and interactive questions.
- First five multiples of 9: 9, 18, 27, 36, 45
- Prime Factorization of 9: 9 = 3 × 3 = 32
| 1. | What Are the Multiples of 9? |
| 2. | First 20 Multiples of 9 |
| 3. | Tips and Tricks |
| 4. | FAQs on Multiples of 9 |
| 5. | Thinking Out of The Box! |
What Are the Multiples of 9?
The multiples of 9 are the numbers which are obtained by multiplying 9 with integers. When we multiply 9 with a positive integer, we get a positive multiple of 9 and when we multiply 9 with a negative integer, we will obtain negative multiples. We don't include fractions when finding multiples. For Example: 9 × 4 = 36
Here, 36 is a multiple of 9. We have learnt that 9 and 4 are called factors of 36. We can also say that 36 is one of the multiples of 4. The other multiples of 4 can be obtained by multiplying 4 with integers.
List of First 20 Multiples of 9
Multiplication is repeated addition. For example, 9 + 9 = 2 × 9 = 18 and 9 + 9 + 9 + 9 = 4 × 9 = 36
Thus, 18 and 36 are the 2nd and 4th multiples of 9 respectively, which can be obtained by adding 9 repeatedly or by simply multiplying 9 with the integers 2 and 4. The other way is to multiply 9 with natural numbers 1, 2, 3, etc. The multiples of 9 are innumerable as there are infinitely many integers. Let's find the first 20 multiples of 9 by multiplying 9 by each of the natural numbers from 1 to 20.
|
Multiply 9 by the numbers from 1 to 20 |
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 |
To understand the concept of finding multiples, let us look at a few more examples.
- Multiples of 6 - The first five multiples of 6 are 6, 12, 18, 24, and 30.
- Multiples of 7 - The first five multiples of 7 are 7, 14, 21, 28, and 35.
- Multiples of 8 - The first five multiples of 8 are 8, 16, 24, 32, and 40.
- Multiples of 10 - The first five multiples of 10 are 10, 20, 30, 40, and 50.
- Multiples of 12 - The first five multiples of 12 are 12, 24, 36, 48, and 60.
- Multiples of 14 - The first five multiples of 14 are 14, 28, 42, 56 and 70.
Tips and Tricks:
- Two numbers that are made up of the same set of digits will have a difference, which is a multiple of 9.
- This property holds true for all numbers made up of the same digits. For example: Consider the numbers 45268 and 86254. Both are made up of the same digits.
86254 - 45268 = 40986 and 40986 = 4554 × 9 which shows that the difference, i.e. 40986, is a multiple of 9.
Think Tank:
- For the pair (9, 36), the LCM is 36. Similarly, the LCM of (12, 36) is 36. Based on this information, can you identify the property which a number and its multiple has?
- What will be the GCF of the above numbers and how is it related to their LCM?
Multiples of 9 Solved Examples
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Example 1: Ms. Cathy wants to arrange 108 children in groups of 9. Is it possible for her to do such an arrangement without leaving out any child? How many groups will be formed here?
Solution:
To check whether any child will be left or not, we need to verify if 108 is divisible by 9 or not.
Sum of digits in 108 = 1 + 0 + 8 = 9, which is a multiple of 9.
Recall: If the sum of all the digits of a number is divisible by 9, then the given number is also divisible by 9. Thus, 108 is divisible by 9.That means, no child will be left if the students are arranged in a group of 9. From the information 9 × 12 = 108.
Hence, there will be 12 groups with 9 students in each group.
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Example 2: Mia and Joe have the same number of cards. Mia arranges her cards in rows of 9 each, whereas Joe arranges his cards in rows of 8 each. What is the minimum number of cards they can have?
Solution:
To get the minimum number of cards, we need to find the least common multiple of 9 and 8. Let's list the first 10 multiples of 9 and 8.
Multiples of 9 = 9, 18, 27, 36, 45, 54, 63, 72, 81, 90
Multiples of 8 = 8, 16, 24, 32, 40, 48, 56, 64, 72, 80We observe that 72 is the number that is a common multiple of 9 and 8. As we continue listing the multiples, we will get many more common multiples. Out of those, 72 is the least common multiple.
Hence, they will have a minimum of 72 cards.
FAQs on Multiples of 9
How do you find the multiples of a number?
We find the multiples of a number by multiplying a given number with integers.
How do you know if a number is a multiple of 9?
A number will be a multiple of 9 if it is completely divisible by 9. In other words, if 9 is a factor of a number, then it is a multiple of 9.
What are the first 4 multiples of 9?
The first 4 multiples of 9 are 9, 18, 27, and 36.
What are the first 10 multiples of 9?
The first 10 multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, 72, 81, and 90.
What is the least common multiple of 9 and 12?
The least common multiples of 9 and 12 can be calculated using the formula: (9 × 12)/GCF(9, 12). Here GCF(9, 12) is the greatest common factors of 9 and 12.
GCF(9, 12) = 3
LCM(9, 12) = (9 × 12)/GCF(9, 12) = 108/3 = 36
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