How to calculate the G.C.D?

There are several algorithms to calculate the G.C.D (Greatest Common Divisor) between two numbers. The easiest and fastest process consists in decomposing each one of the numbers in products of prime factors, this is, and we successively divide each one of the numbers by prime numbers till we reach a quotient that equals 1. I’m going to give you an example so as to be easier for you to understand. We want to calculate the G.C.D between 168 and 180. We start by factorizing each one of the numbers.

Factorizing enables us to arrive to the conclusion that `168 = 2^3 xx 3 xx 7` and that `180 = 2^2 xx 3^2 xx5`. The following step is getting the product of common factor with a smaller exponent: `2^2 xx 3 = 12`. So, we can conclude the Greatest Common Divisor between 168 and 180 equals 12.

What is the G.C.D useful for?

There are several problems where the calculation of the G.C.D is useful. Let’s suppose that a florist has 168 roses and 180 daisies and she wants to form the more bunches she can having both types of flowers and having the same amount of flowers. In this situation, knowing that the G.C.D is 12 it is enough to do `168:12 = 14` and `180 : 12 = 15`. Thus, it is possible to form 12 bunches having each one 14 roses and 15 daisies.

Can we use the G.C.D to make the fraction irreducible?

Yes, you can. It is also useful to do it. Supposing we have a fraction made up by the previous numbers, it is enough to divide the numerator and the denominator by the G.C.D in order to make that fraction irreducible. Thus`168/180 = 14/15`

But, after all, what is the G.C.D of two numbers?

First of all, I want to stress that the G.C.D is not only calculated between two numbers. We can calculate the G.C.D of 2, 3, 4 or more numbers! Thus, if we have two natural numbers we assure you that there will be common divisors among them. If there is only a common divisor is because that divisor corresponds to number 1 and in that situation they are called relatively prime numbers. But if they have several common divisors, so the G.C.D is the biggest of those divisors.

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