Factoring - Greatest Common Divisor

One final fun fact regarding the Greatest Common Divisor is demonstrated in the following question:

Which of the following is the greatest common divisor of 6 and 12?

Incorrect. 1 IS a common divisor of 6 and 12, but it is not the Greatest common divisor.

Incorrect. 2 IS a common divisor of 6 and 12, but it is not the Greatest common divisor.

To sum up:

[[summary]]

Incorrect. 3 IS a common divisor of 6 and 12, but it is not the Greatest common divisor.

Incorrect. 6 is not divisible by 12, so 12 cannot be a common divisor of 6 and 12.

Common divisors - definition: A common divisor of two integers is an integer that is a divisor of both. For example, 2 is a common divisor of 6 and 8, because both 6 and 8 are divisible by 2.

All integers are divisible by 1, so 1 is always a common divisor of any two integers. However, some GMAT integers questions ask about the Greatest Common Divisor (GCD) of two or more integers, and whether the GCD is greater than 1 or not. In the above case, 2 is also the Greatest Common Divisor of 6 and 8.

Since prime numbers are divisible only by themselves and 1, any two primes will only have a GCD of 1. However, note that the reverse isn't necessarily true: The fact that two integers have a GCD of 1 does not necessarily indicate that they are Primes. This is a common misconception, easily disproved by example: 8 and 9 have a GCD of 1, even though neither of them is Prime.

Correct. Since both 6 and 12 are divisible by 6, 6 is a common divisor of both 6 and 12. And since 6 cannot be divisible by any number greater than itself, it is also the Greatest Common Divisor of 6 and 12.

Remember that any integer is always a factor of itself, and therefore any integer could serve as the Greatest Common Divisor of itself and another integer.