Greatest Common Divisor

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Formal Definition

Let r and s be two positive integers. The positive generator d of the cyclic group $H=\{nr+ms|n,m\in\mathbb{Z}\}$ under addition is the greatest common divisor (abbreviated gcd) of r and s. We write $d=\gcd(r,s)$.

Informal Definition

The greatest common divisor (or GCD) between two numbers is a number that is both a common factor between the them and also the largest of the common factors.


The greatest common divisor of 42 and 72 is 6.


  • The GCD of 50 and 25 is not 5. Even though 5 is a common factor between the two integers, it is not the largest.

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