Another way to see or understand the Greatest Common Divisor will be defining it as the largest number that divides two or more natural or whole numbers.
For example, if we divide 10 by 5, the result will be 2 and the remainder equal to zero. But if we divide 10 by 6, the closest value is 1 and the rest would be 4; therefore, we will understand that 6 is not a divisor of 10, since its remainder is not equal to zero.
Thus, successively, we will go testing with the numbers, to find the divisors and then determine what the greatest common divisor will be. Let’s say that this would only be a method and a first step to carry out this operation, there are others that we will explain laterTo denote the greatest common divisor, we must write: GCD (a, b). The “a” and “b” are the numbers to which we will calculate the greatest common divisor. For example, if we want to calculate it at 15 and 20, we would denote it as follows: GCD (15, 20).
Before continuing, we must clarify that this mathematical operation can only be performed with whole numbers. With decimals it is not allowed, as it would not work.
Ways to calculate the Greatest Common Divisor
There are two ways. If you want to find the Greatest Common Divisor of small numbers, you can use the form 1, which is simply to find the divisors of two or more numbers and see which of all is the maximum that divides them exactly. But when it comes to larger numbers, it is best to use form 2, which refers to factoring.
Form 1: Calculating the divisors.
For instance. If we want to find the greatest common divisor of 10 and 20.
First, we must write one below the other and place their dividing numbers. As follows:
10 = 1, 2, 5, 10.
20 = 1, 2, 4, 5, 10, 20.
How do you find the divisors? Easy, you divide the number (10, for example) from 1 onwards. Those where the remainder is equal to zero, then they will be divisors of 10.
Second, once you find the divisors of 10 and 20 (separately), you must identify which ones match or divide both numbers.
In this case, the matching divisors are 1, 2, 5, and 10.
As the greatest common divisor, it is the largest number that divides two or more numbers, this time, the greatest common divisor of 10 and 20 will be 10.
Dimensioning : This is a simple and logical way to understand how to find the greatest common divisor of one or more numbers. However, it is considered the long way to perform such a calculation. Once we understand the logic of operation of said calculation, we always proceed to do it through the decomposition of factors. This other way is much more practical and shorter, and it also works for larger numbers.
Form 2: Decomposition of Factors.
This way consists of decomposing into prime factors (separately) the numbers for which you want to calculate the greatest common divisor and, then, through the numbers that coincide, choosing the largest number that divides both. As follows:
First, the decomposition always begins by placing the number (to be decomposed, in this case 40 first) and drawing a vertical line next to it, to place (on the other side) its prime factors.
Second, the decomposition always begins with the smallest divisor, that is, the smallest number that can divide the number we are analyzing. In this case, it would be number 2. And so on until finished.
Third. We already know, then, that the divisors of 40 are 2x2x2x5.
Quarter. Now, we proceed to do the same (decompose) with the 60.
In the case of 60, the divisors are 2x2x3x5.
Fifth. From the divisors of both numbers, we are going to choose the ones that repeat. This time they are: 2x2x5.
Sixth. We proceed to multiply them. The result would be 20. Therefore, the greatest common factor between 40 and 60 is 20.