Greatest common divisor

The greatest common divisor or GCF is very useful in the world of mathematics by allowing us to simplify fractions quickly and easily.

What is the greatest common divisor?

It is known as the  greatest common divisor, abbreviated GCD,  of two, three or more numbers , the largest whole number  , common to all of them, which allow dividing them without leaving a remainder . In a more technical concept, it is the greatest common divisor with respect to all those numbers.

  • Definition of greatest common divisor
  • What is the greatest common divisor for?
  • How to find the greatest common divisor
  • Example exercises and problems

Definition of greatest common divisor

Within existing concepts in mathematics , is  defined as the  greatest common divisor  (MCD) of  two or more  integers  greater  integer that divides without leaving  any residue number

What is the greatest common divisor for?

The GCF is used to simplify to fractions . If, for example, to simplify a fraction, first calculate the GCD (60, 48) = 12, where both the numerator and the denominator of the initial fraction are divided by 12 to obtain the fraction.

The GCF can also be used to find the  least common multiple of two numbers . Precisely, the product of the two numbers is equal to the product of their greatest common divisor by their least common multiple . Likewise, to calculate the least common multiple of 48 and 60, we first calculate their GCF, 12, being their least common.

In other aspects of mathematics  more  complex the MCD is used and the  algorithm of Euclides are used in the resolution of  Diophantine equations linear with two incógnitas. El algorithm Euclide s is used in the development of a rational number in the continued fraction.

How to find the greatest common divisor

There are two methods to calculate the greatest common divisor , these are:

  • The simplest and most direct method is to extract from all the numbers that are established and their corresponding divisors . The highest divisor that is repeated in all the questioned numbers is the GCF For example: GCF (20, 10)
    • Those that divide 20: 1, 2, 4, 5, 10 and 20.
    • Those that divide the 10: 1, 2, 5 and 10.
    • The highest divisor common to both is 10 , and that is why its GCF is the number 10. This method is valid for small numbers , because it is simple , but if it is  high numbers it will be more complicated , that is why that there is a second method.
  • The second method is done through  factor decomposition , it is the most common and used method. It is decomposed cad a number who ask us in all its divisors. Once this is done, we have to take all the common factors with the lowest exponent and multiply them between them.

Example exercises and problems

Below is a list of problems to solve :

  • Juan and Rafael eat at the same restaurant, but Juan attends every 32 days and Rafael every 14. When will they meet again?
  • Maria has 24 pieces of candy to distribute and Alan has 16. If you want to give the candy to your family group so that everyone has the same amount and is as large as possible, how many pieces of candy will you distribute to each person? How many family members will each of them give candy to?
  • Sami has a 220 meter cable and a 48 meter cable. You want to cut them so that all the pieces are equal but as long as possible. How many pieces of wire will you get?
  • In one neighborhood, a hot dog’s cart passes every 14 days and an ice cream truck passes every two weeks. It is known that 12 days ago both vehicles passed on the same day. Billy believes that in a month and a half the vehicles will meet again and Ciro believes this will happen in two weeks. Who is correct?
  • In a religious worship group which is composed of a pianist, a bassist, a guitarist and a drummer, the drummer plays in 6-beat lapses, the guitarist in 8-beat, the bassist in 10-beat, and the saxophonist in 12-beat. If they all started at the same time, how many times would your periods start over at the same time?
  • John has a racetrack with two model cars. The first car makes a complete lap around the track in 18 seconds while the second does it in 30 seconds. Thomas also has his two-car racing circuit, but the first makes a full lap in 32 seconds and the second in 38 seconds. As Thomas always loses when they play, he proposes to John that the winner is whoever has both of his cars on his track at the finish line at the same time. Who will win?
  • Piero wants to paint his house, which is small. According to my calculations, I need about 14 liters of white paint, 16 liters of black paint and 24 liters of gold paint. But you want to buy cans of paint that have the same number of liters and that the number of cans is as low as possible, how many liters should each can be and how many cans of each color should Maximum buy?

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