In mathematics, the greatest common divisor (gcd) is the largest possible positive integer which divides the numbers with zero remainder. Greatest common divisor is also known as which is also known as the greatest common denominator, greatest common factor (gcf), or highest common factor (hcf).
For example, the GCD of 14 and 63 will be 7.
In the name "greatest common divisor", the adjective "greatest" can easily be replaced by "highest", and the word "divisor" also can be replaced by "factor", so that’s why we also known greatest common divisor as Highest common factor also these are called as the greatest common factor (gcf), etc.
The GCD of two or more integers will be the largest integer which will divide each of the integers such that their remainder will be zero.
GCD of 20 and 30 = 10 (As 10 is the highest common number which divides 20 and 30 with remainder as zero).
GCD of 42, 120 and 285 = 3 (3 is the highest common number which divides 42, 120 and 285 with remainder as zero).
How to Calculate the G.C.D?
There are various methods or algorithms to determine the G.C.D (Greatest Common Divisor) between two given numbers. If we talk about the easiest and fastest process so it consists in decomposing each one of the numbers in the products of prime factors, this is, and then we successively divide each one of the numbers by the prime numbers until we reach a quotient that equals 1.
We are going to discuss with you an example so as to be easier for you to understand. We want to determine the G.C.D between 168 and 180. First of all We start by factoring each one of the numbers as shown below.
By doing Factors we shall arrive at the conclusion that 168 = 2 × 2 × 2 × 3 × 7 and that 180 = 2 × 2 × 3 × 3 × 5.
In the next step we shall determine the product of common factors with a smaller exponent: 2 × 2 × 3 = 12.
Finally, we can conclude the Greatest Common Divisor between 168 and 180 will be equal 12.
Applications of Greatest Common Divisor
There are various problems in which the determination of the G.C.D is very useful. Let’s assume that a florist has 180 roses and 168 daisies and she wants to make the count of bunches in which she can have both types of flowers (roses and daisies) and having the same amount of flowers. In this situation, by determinin that the G.C.D is 12 it is enough to do 168:12=14 and 180:12=15. Thus, it is possible to make 12 bunches which will be having each one with 14 roses and 15 daisies.
Applications of LCM and GCD Definitely Help in Quite a Lot of Things.
Here are some of them:
Helps in arithmetic for solving fractions. Doesn't it? Fractions with different denominators, to solve them we first bring them to have common denominators.
Helps to find commonality. 2*4=8, 4*2=8 as well, we can use either 2 four times or 4 two times to bring a eight. Cases where you need to distribute, what if the same 10 chocolates were to be given to five kids instead of 2?
We usually find use of GCD in measurements and construction fields.
If we know the LCM and GCD, then we can simply find the product of numbers.
What is the G.C.D of Two Numbers?
First of all, we have already discussed that the G.C.D is not only calculated between two numbers. We can determine the G.C.D of 2, 3, 4 or more numbers as well Thus, Let us assume we have two different natural numbers Now we assure you that there will be common divisors among them. If there is a case in which only a common divisor is because that divisor corresponds to number 1 and in that situation these are called relatively primary numbers. But if they have various common divisors, so the G.C.D will be the greatest of those divisors.