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# Greatest common divisor - gcd

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## What is gcd?

Of the divisors that have several natural numbers in common, the greatest of these divisors is known as the greatest common divisor, also abbreviated as gcf. In other words, it is the largest number that divides exactly two or more numbers.

Note: If you want to obtain the LCD of a single number, the LCD will correspond to the same number. For example, the divisors of 15 are 1, 3, 5 and 15; Therefore, the LCD is 15.

## How to get the greatest common divisor?

To obtain the gcf it is advisable to draw a vertical line where, to the left of the line are the numbers from which we want to obtain the greatest common divisor.

To the right of the line we will put the prime number that will divide the number from which we want to obtain the gcf.

We are going to make an example for which we use the same procedure for 2 or more numbers, from which we want to obtain the greatest common multiple. Considering as an example to obtain the gcf of 20 and 30.

1. First we must draw a vertical line and put a separation between the numbers to be analyzed, in this case it is 20 and 30. It is very important to have an order.

20    30

Note: A line was drawn to visualize each stage of the operations.

2. It is checked if the numbers to be analyzed are divisible by 2, we observe that the two numbers 20 and 30 can be divided by 2, it is important to consider that to obtain the gcf the two numbers must be divisible and the result of the division must be integer ( no residue), therefore, the number 2 is placed to the right of the line and to the left of the line in the next row the result of the division is placed.

20    30  2

10    15

3. Again we must check if the two numbers, now 10 and 15, are divisible by 2, as we can see 10 if it is divisible by 2 but 15 is not divisible by 2, therefore it is not possible to use the prime number 2. Since it is not possible to divide the two numbers by 2 we pass to the next prime number that is 3, but it is not possible to divide the two numbers by 3 either (The number 15 if it is possible to divide by 3 without having a residue but the number 10 is not possible divide by 3 since you would have a result of 3.3333), therefore, we continue to the next prime number that is 5. As we can see both 10 and 15 are divisible by 5.

20    30  2

10    15  5

2      3

4. When the results of the numbers to be analyzed (in this case 2 and 3) it is no longer possible to divide by the smallest prime number, in this case 2 (Since only 2 is divisible by 2), therefore, it is already possible get the gcf by multiplying the numbers to the right of the vertical line.
2 x 5 = 10
The gcf of 20 and 30 is 10.

Examples:

A)

8    6  2

4    3

B)

15    18  3

5      6

C)

24    33  3

8    11

D)

42    18  2

21      9  3

7      3

The gcf is:
A) 2
B) 3
C) 3
D) 2 x 3 = 6

Exercise:

What is the greatest common divisor? gcd =?
A)

6    26    = ?

B)

14    21    = ?

C)

24    16    = ?

D)

45    25    = ?

## Calculation of the gcf in a fraction

Assuming we have the following fraction:

18 / 22

We want to simplify the fraction or division, therefore, we must obtain the gcf of 18 and 22.

First we must arrange the numbers for which it is required to obtain the gcf, in this case it is 18 and 22, following the procedure to obtain the greatest common divisor the result was as follows:

18   22  2

9   11

As you can see, 11 is a prime number, but 9 between 11 is not an integer, therefore, the gcf of 18 and 22 is 2.

Now that we know the gcf the fraction of simplifies:

18 / 22 = 9 / 11

We can check when dividing: 18/22 = 0.818181 y 9/11 = 0.818181.

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