# How To Find Tne Multiples (Factors) And The Divisors Of A Whole Number

In this lesson, we will learn how to determine the multiples and divisors of a natural integer, and the relation between them. An easy and important lesson for studying arithmetic lessons. We hope you benefit from our lesson.

## Divisors Of A Natural Integer

Let's take the number 24, for example. If we divide the number 24 by the numbers from 1 to 24, we will notice that the result is always either a natural integer (the remainder is equal to 0) or a non natural number (meaning the remainder is different from 0).

Let's do this division:

So the divisors of 24 are the following numbers: , and is not divisible by the others.

• The number 24 is called: Dividend
• The number 8 is called: Divisor
• The number 3 is called: Quotient

So this property can be summarized as follows:

### Note:

Let , and are three whole numbers with is non null. The number is a divisor of the number if the remainder is equal to 0.

If the number accepts only two divisors, 1 and itself, we say that the number is a prime number.

#### Example1:

Find the divisors of each of the following numbers: 27; 29; 48; 72 and 100.

#### Solution:

The divisors of 27 are:

The divisors of 29 are: , so, 29 is a prime number.

The divisors of 48 are:

The divisors of 72 are:

The divisors of 100 are:

## Multiples Of A Natural Integer

For example, let's take the number 8 and determine its multiples. To find multiples of 8, we multiply 8 by 1, then by 2, then by 3, and so on. We get an infinite set of multiples.

To be more clear, we have:

So as you can see, the multiples of 8 are the set: , where we get the current multiplier by adding 8 to the multiplier before it.

• The number 8 is called: Factor 1
• The number 3 is called: Factor 2
• The number 24 is the multiple of the two factors 3 and 8.

So this property can be summarized as follows:

### Note:

Let , and three natural integers.

is the multiple of if :

## Special Cases Of Divisors And Multiples

• 1 has only one divisor that is itself
• The divisors of 0 are all whole numbers except 0.
• The multiples of 1 are all whole numbers.
• 0 has only one multiple that is itself.

#### Example2:

Find the multiples less than 60 of each of the following numbers: 6; 7; 9; 11 and 20.

#### Solution:

The multiples less than 60 of 6 are:

The multiples less than 60 of 7 are:

The multiples less than 60 of 9 are:

The multiples less than 60 of 11 are:

The multiples less than 60 of 20 are:

## The Relation between Divisors And Multiples

Let's go back to the example we took in the divisors section. The divisors of 24 are: .

And the example we took in the multiples section. The multiples of 8 are: .

We notice that the number 8 is a divisor of 24, and the number 24 is a multiple of 8.

### Note:

If is a divisor of , then, is a multiple of .

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