Let's Look At The Greatest Common Divisor Of Two Numbers?
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
- 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.