Multiples of 3 and How to Find Them

When we talk about Multiples of 3 it is the set of all those numbers which can be expressed in the form of 3n or in other words the number which gives the remainder zero when it is divided by 3 is called a multiple of three. All the multiples of three must contain 3 as it’s one of the factors.

For example, number 18 can be expressed as 2 × 3 × 3 thus we can see that 3 is one of its factors also it is expressed in terms of 3n so this is a multiple of three. Thus, in other words, we can say that all those numbers can be divided by three or are products of 3, and any number is defined as a multiple of 3.

Multiples of 3

Here's an article explaining various terms related to multiples. In this article, you are going to learn - what is a multiple, what are the different Multiples of 3, how to find the Multiples of 3, some important FAQs, etc. At the end of this article, you will be able to know how to find multiples of different numbers as we will be providing you with some examples with the help of which, you will easily be able to understand the topic. 

What is Multiple? 

When we multiply a number (whose multiple, we have to find) with a positive integer, the result we get is multiple. For example, if we have to find the multiple of the number 2, we can multiply by any other number, say 3, we will get 6 as an answer. Here the number 6 is the multiple of 2.

We can use a simple formula to check the multiples of a number. This formula is as follows:

Multiple of X = Xn ( 'X' is the number whose multiple we have to find and n is any positive integer.)

So, the multiple of a number can be defined as the number that can be written as the product of a given number and some other natural number. Multiples of the numbers can be observed in the multiplication table. Multiples of natural numbers are as given below:

E.g., multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, and so on. Hence, multiples of 2 will be even numbers and will end with 0, 2, 4, 6, or 8.

The multiples of the number 3 are 3, 6, 9, 12, 15, 18, 21,24,.............. and so on.

The multiples of the number 5 are 5, 10, 15, 20, 25,.............., and so on.

All the multiples of 5 will have the last digit as 0 or 5.

From the above-given examples we can say that multiples of 2, the number 2 can be multiplied by infinite numbers to find the “n” number of multiples.

Calculation of Multiples of 3

The multiples of the number 3 can be calculated by multiplying integers. For example, to calculate the Multiples of 3 we will use the product of 3 with the natural numbers 1, 2, 3, .......... and thus will get 3 x 1, 3 x 2, 3 x 3, 3 x 4, 3 x 5, etc., which equal 3, 6, 9, 12, 15, etc. All the Multiples of 3 that come in the table of three are Multiples of 3 i.e. 3, 6, 9, 12, 15, 18, 21, 24, 27, 30……………….etc. Thus Multiples of 3 are expressed as 3p where p is an integer.

Example:

To find the Multiples of 3, we have to multiply them with many numbers. We will start by multiplying it with 1 and will end by multiplying it with 10. This will give us the first 10 Multiples of 3.

i.e.,

3 × 1 = 3

3 × 2 = 6

3 × 3 = 9

3 × 4 = 12

3 × 5 = 15

3 × 6 = 18

3 × 7 = 21

3 × 8 = 24 

3 × 9 = 27

3 × 10 = 30

So, the first 10 Multiples of 3 from above come out to be 3, 6, 9, 12, 15, 18, 21, 24, 27, 30. 

Similarly, we can find more multiples of the number 3 by multiplying them with more natural numbers. 

Let's solve some more examples which will help you to understand this topic easily. 

Example 1: Find the first 15 multiples of 7.

Solution: To find the first 15 multiples of 7, we will multiply it by the positive integers from 1 - 15.

So, the multiples are:

7 × 1 = 7

7 × 2 = 14

7 × 3 = 21

7 × 4 = 28

7 × 5 = 35

7 × 6 = 42

7 × 7 = 49

7 × 8 = 56

7 × 9 = 63

7 × 10 = 70

7 × 11 = 77

7 × 12 = 84

7 × 13 = 91

7 × 14 = 98

7 × 15 = 105

Therefore, the first 15 multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105. 

Example 2: Find the first 5 multiples of -3.

Solution: The first 5 multiples of -3 are as follows:

-3 × 1 = - 3

-3 × 2 = - 6

-3 × 3 = - 9

-3 × 4 = - 12

-3 × 5 = - 15

Hence, the first 5 multiples of -3 are -3, -6, -9, -12, -15.

Set of Multiples of 3

Multiples of 3 can be written in the form set in the roster or tabular form as well as in the inset builder.

1. In roster or tabular form, the numbers of the set are written within brackets separated by the commas so multiples of three will be written as

{0, 3, 6, 9, 12,...}

2. Inset Builder form elements of a set are written with their properties thus the Multiples of 3 will be written as

{x:x = 3n, n ∈ W, where W is the whole number}

Difference between Factors and Multiples

The exact divisors of the given number can be defined as the factor of the given number while the multiples of the number are defined as the numbers obtained when multiplied by other numbers.

The number of factors of any number is always finite while the number of multiplies of the number is infinite.

The operation which is used to find the factors of any number is a division while the operation used to find the multiples of a given number is called multiplication.

The result or the outcome of the factors in any given number will always be less than or equal to the given number while the result or the outcome of the multiples should be greater than or equal to the given number.

Now, let us assume an example:

3 × 4 = 12

Here, 3 and 4 are the factors of 12,

12 is multiple of 3 and 4

Thus, we can conclude that if X and Y are two numbers and:

X is a factor of Y if  X divides Y.

Y is a multiple of X if Y is divisible by X.

We know that the number 1 divides every integer thus it is the common factor of every integer, and also every number is divisible by 1 and every number is a multiple of 1.