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What Is Greatest Common Divisor?

In Factorization, the Greatest Common Factor (GCF) is also known as Greatest Common Divisor (GCD).

So what does it mean? The simple answer is a common factor of two numbers is any number that is a factor of both. In other words, both numbers can be divided evenly by the same number. For Example the number 6 is a common factor of 108 and 126.

The greatest common factor meaning is more specific than common factor; it means the largest value of all the common factors. Example, the common factors of 108 and 106 are 1, 2, 3, 6, 9, and 18. In this case the greatest common factor is 18 because it is the largest number of the common factors. Before going further, here is recommend reading: Exponents, and Prime Factorization.

How do we find the greatest common factor / greatest common divisor of any two numbers? First find the Greatest Common Factor of both numbers; do this by factoring both numbers into their primes using Prime Factorization. Secondly, find all the primes they have in common and multiply them together. Example, we know the Greatest Common Factor of 108 and 126 is 18. Here is how to find the answer 18:

• Prime Factorization of 108 = 2 x 2 x 3 x 3 x 3 = 2² x 3³
  
  
• Prime Factorization of 126 = 2 x 3 x 3 x 7 = 2 x 3² x 7
  
  
• Greatest Common Factor / Greatest Common Divisor: is 2 x 3 x 3 = 2 x 3² = 18.
  How do we get the answer 18? The answer is to take the common numbers of the Prime Factorization of 108 and 126 and multiply them together. Notice 2¹ is less than 2², 3² is less than 3³, and 7 is excluded because it is not common to both numbers. These multiplied together to produce the GCF / GCD 18.
  
  
• (a) We see 18 divides evenly into 108 six times: 108 ÷ 18 = 6
  (b) We see 18 divides evenly into 126 seven times: 126 ÷ 18 = 7
  (c) Note both numbers divide evenly which is our goal.
  
  

There is a important exception. See Relative Primes.