29 November 2023 When we talk about mathematics, few like it and the reason is that sometimes it is difficult for us to understand its concepts perfectly. However, this science, let’s say that it is the mother of all the others and has an important function through time to tune our brain. It helps us in logical reasoning, and believe it or not, it is essential to organize the chaos in our minds.

## Definition and concept of the greatest common divisor

In a simple way, the definition of the greatest common divisor (GCD) is that of that digit or greater number, which allows sectioning several numbers. The divisor would be that amount that dividing by another gives a remainder of zero. So that we understand better, let’s see an example: 15÷3 gives a remainder of 0 when doing the division.

Which gives us the reference that the greatest common divisor (GCD) is the digit that gives zero as a remainder when dividing different numbers together. This makes it become the maximum factor and the common word comes from the quality of being able to be the divisor of a group that contains several numbers.

## What are the uses of the greatest common factor?

The greatest common factor is commonly used to find common denominators. The denominators are those numbers in the fractions that indicate how many portions the unit is divided into. To identify it, we look at the bottom of the fraction. But now, when talking about the common denominators, reference is made to the number that is in that position and it will be the same for all the fractions of the mathematical exercise.

## What are the properties that a greatest common divisor must satisfy?

Among the properties of the MCD we can highlight the following:

• Yeah divide or multiply two numbers by a third digit, its greatest common divisor will also be divided by this third digit. For example, we have 15 and 20. The LCD of both is 5. Now, if we multiply this by 3 including the LCD, we get the following: 15×3= 45, 20×3= 60 and its LCD 3×5= fifteen.
• To the Divide two digits by their GCD, The result of the divisions will be in prime numbers. A prime number is one that can be divided by itself and by 1. These are given in natural numbers, those that do not comply with this rule are known as compounds. As an example of a prime number we have 2, 3, 5, 7, 11, among others. To exemplify this property you can divide 735÷5 and 550÷5. You will obtain, respectively, 147 and, on the other hand, 110. Now you will notice that when performing the decomposition of each result, what is given is 1 in both. With which the determination is reached that they are cousins.

When one number can divide another, it is said that is the greatest common divisor of both. What is stated is seen in the following:

• It is known that 4 is a divisor of 24. So, if the calculation of the GCD is made of what will be obtained, must be 4, according to this property.

## How is the greatest common factor calculated?

There are several ways to calculate the greatest common factor. But there are three methods that are distinguished by their general use:

• The prime decomposition: Here the prime numbers are decomposed into factors. Think about what you want to know is the greatest common factor of 8 and 12.

For it you must factor:

• 8/2=4
• 4/2=2
• 2/2=1
• 12/2=6
• 6/2=3
• 3/3=1
• 8=2³
• 12= 2²×3
• 2²=4

In this method, factors that are common are taken into account. In the example, 2 is common. However, you must choose the one with the best exponent. Here was the 2² that would be equal to 4.

There is also the way to get the greatest common divisor by Euclid’s algorithm. This is one of the oldest methods for calculating the greatest common factor. The author was an ancient Greek mathematician This is known as the father of geometry and he developed a series of steps to find the greatest common divisor of two figures.

The algorithm consists of dividing the largest digit by the smallest. In the event that the division is exact, the divisor is assumed to be the GCD. In the fact that the operation is inaccurate, continue to divide the divisor by the remainder until it is exact. The GCD will be the last quantity that fragments, now we will look for the GCD of 138 and 42.

First of all, the two numbers must be divided the greater among the lesser.

Subsequently, it is done the multiplication of 42 for the entire part of the fruit of the previous operation.

Then proceed to perform the subtraction of 138 minus the end of the multiplication 126.

After this, 42÷12=3.5 is divided, but the one that is obtained is still not integer. For what corresponds to continue in the search of the MCD. To do this, we perform the multiplication of 12 by the integer part of the previous division. 12×3=36

• Next, subtract 42 minus 36, which gives you 6.

We do the fraction again, this time, 12÷6= 2. Here we already obtained the MCD because the result was an integer, and it is the last divisor, 6.

### Greatest common divisor of two numbers

The greatest common factor is the greatest factor that divides a group of numbers. We can obtain the greatest common divisor of two digits or more than two numbers, That will depend on the steps to follow for that. In the case of the GCD of two numbers, it is known that both a and b have a greater number that divides them, let’s see the following example.

We are asked to calculate the greatest common divisor of 12 and 18, since we are looking for their greatest common divisor, the numbers cannot be greater than 12. So we have the following: 1,2,3,6, they are the elected candidates, since they must divide the two digits. In such a case, the greatest common divisor of these is 6, because it is the greatest of all the options.

### Greatest Common Divisor of Polynomials

When we refer to the greatest common divisor of two polynomials, it is necessary that they be other polynomials. This must be of the highest degree possible, since it would be the factor of two initial polynomials. We can compare what the GCD of polynomials means with the greatest common divisor of integer figures, that is, it is similar.

The most recommended way to find the GCD of polynomials It is through factoring. What is sought by this means is to find the repetitive factor between the polynomials.

### LCM of a fraction

The LCM is the smallest number that can be multiplied in common by two or more digits. To understand it better, let’s look at the multiples 6 and 8, so we are going to multiply in the case of 6 by 1, 2, 3,4 and so on. In the same way, proceed with 8.

• 2×1=2
• 2×2=4
• 2×3=6
• 2×4=8

These would be the multiples of 2, the results of the different multiplications. Let’s see with 8.

• 8×1=8
• 8×2=16
• 8×3=24
• 8×4=32

And so we see from one appearance simple multiples of 8.

Now, in the case of fractions, the LCM, the way to get the smallest number changes something. They recommend that the first thing to do is to identify the denominators. Then separate them a bit and draw a vertical line to the right of each one. What follows is to select the prime number that comes after 1, that is, 2, with which you will begin to divide by denominator.

If either is not divisible by 2, do a small horizontal line and go to the next prime number. That runs until it’s divisible, but if they can all be divided by 2, then you do the operation and write the result you got.

See if the columns that gave you a result of 1 are already ready. You must continue then for those that were not possible to be divided by 2. Therefore, it goes to the next prime number, which is 3, if it does not work with 3, it happens to 5.

Already when all the columns formed by the lines drawn at the beginning result in 1, it is confirmed that it is finished. Remember that the first column does not count, because it contains the numbers to be used to divide the initials.

The least common multiple is the data used in the division, that is, it is the figures that are in the column that is first. The fractions to calculate the least common multiple:

The greatest common multiple of 9,5,10,4 is 2²×3²×5 Such an operation would result in 180.

When numbers have powers the GCD is calculated as follows:

• Each number is broken down into prime factors and factors common to both numbers are identified.
• The common factors with the least power in each number are taken and multiplied to obtain the LCD.

For example, if we want find the GCD of 24 and 36, First we break them down into prime factors:

• 24 = 2^3 x 3^1 36 = 2^2 x 3^2

The factors common to both numbers are 2 and 3. We take the common factors with the least power in each number: 2^2 and 3^1. So the LCD of 24 and 36 is 2^2 x 3^1 = 12.

## Examples of the greatest common factor in mathematics

Below are simple examples of the greatest common factor in mathematics.

Calculate the GCD of 6, 12, 15

Begin by dividing each number by the first prime number that follows 1, that is, 2. If it is divisible, use 2, otherwise you pass to the prime number that follows it, for example 3. The important thing is to find the number that can divide the results until reaching 1, that must be the end, we can exemplify it like this:

• 6/2=3
• 3/3= 1 the division of 6 we leave it here.

Now we go to 12:

And finally the 15th:

Therefore, 6= 2×3 12=²×3 15=3×5

Which means that the GCD of 6, 12 and 15 is 3 because it is the divisor that is repeated in all three cases. In the example, the greatest common divisor of this set of numbers is 3. Since it is the largest number, it is repeated in the division using the least common multiple. This is a very practical technique that is useful when finding the MCD without getting so confused.