﻿ Divisor — Math18

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# Divisor

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## What is a divisor?

A divisor is an arithmetic function that divides the natural number into exact parts, therefore, dividing the quotient must result in an integer (this means that it does not contain a decimal point) and the remainder or remainder of the division is 0. It is Possible to represent the divisor of the number as follows:

n = main number / divisors

Where "n" is any natural number.

Mathematically, the representation to identify the multiples of a number is by means of the letter “D” in uppercase and then in parentheses the main number from which we want to obtain the numbers that are divisors of it, we equate to the divisors that are enclosed in keys of as follows:

D (main number) = {Dividers}

For example, the divisors of 8 are represented as follows:

D(8) = {1, 2, 4, 8}

The dividers of 8 are obtained when performing the operations 8/1 = 8, 8/2 = 4, 8/4 =2 y 8/8 = 1.

## How to know if a number is a divisor?

To know if a number is a divisor of the main number, the best way is to perform the division and verify that the quotient is an integer, that means that the remainder or remainder of dividing the “main number / number to be known” must be equal to zero.

n = main number / number to be known

Where "n" is any natural number.

## Properties and characteristics of the dividers

• Every divisor of a principal number will always be less than him.
• The number 1 is a divisor of all numbers that are greater than 0.
• Every major number will be a divisor of itself, giving the quotient as 1.
• Every prime number is only divisible by one and himself.

## Finding Dividers

An alternative to find the divisors is to divide the main number between the prime numbers until the last number is 1.

Suppose we want to find the divisors of 60, first a vertical line is drawn and to the left the number from which the divisors are to be obtained is placed, to the right the dividing numbers are placed.

60  2 30  2 15  3   5  5   1

The previous method is done in the same way to obtain the least common multiple, if you have any doubts about the procedures you can see the link of least common multiple (lcm) so that you acquire more knowledge.

Now we are going to organize the divisors obtained in a table, in this case we observe that the number 2 is repeated 2 times therefore it must be multiplied 2 x 2 = 4 to add to the column and row part that they are going to multiply.

Note: You must always add the number 1.

 X 1 2 4 3 5 1 1 2 4 3 5 2 2 4 8 6 10 4 4 8 16 12 20 3 3 6 12 9 15 5 5 10 20 15 25

You should check with each different number in the table and verify if it is possible to divide by 60.

It has:

60/1 = 60
60/2 = 30
60/4 = 15
60/3 = 20
60/5 = 12
60/8 = 7.5 (It is not divisor)
60/6 = 10
60/10 = 6
60/16 = 3.75 (It is not divisor)
60/12 = 5
60/20 = 3
60/9 = 6.6667 (It is not divisor)
60/15 = 4
60/25 = 2.4 (It is not divisor)

From the previous results it is checked which numbers can divide 60 into an integer, it is also important to consider the result as a dividing number, for example: 60/2 = 30 it is also possible 60/30 = 2, therefore, 2 and 30 are divisors of 60.

As a result, the divisors of 60 are:

D(60) = {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 y 60}

D(2) = {1, 2}

D(3) = {1, 3}

D(4) = {1, 2, 4}

D(13) = {1, 13}

### Divisors of 14

D(14) = {1, 2, 7, 14}

### Divisors of 18

D(18) = {1, 2, 3, 6, 9, 18}

### Divisors of 27

D(27) = {1, 3, 9, 27}

### Divisors of 28

D(28) = {1, 2, 4, 7, 14, 28}

### Divisors of 54

D(54) = {1, 2, 3, 6, 9, 18, 27, 54}

## Divisors of two or more numbers at once

To obtain the common divisor of two or more numbers it is important to know how to obtain the greatest common divisor (gcf).

Assuming we want to find the divisors of 4 and 12, following the procedure to obtain the gcf you get:

4   12  2 2     6  2 1     3

Therefore, the gcf of 4 and 12 is:

2 x 2 = 4

When plotting the multiplication table we find the divisors of 4 and 12.

 X 1 2 4 1 1 2 4 2 2 4 8 4 4 8 16

This means that the divisors of 4 and 12 are:

D(4, 12) = {1, 2, 4}

Important: Divisors must not exceed the value of the gcf.

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