Brent@GMATPrepNow wrote:x and y are positive integers. If the greatest common divisor of 2x and 2y is 30, what is the greatest common divisor of x and 2y?
1) y is odd
2) x is odd
Target question: What is the greatest common divisor of x and 2y?
Given: the greatest common divisor of 2x and 2y is 30
30 = (2)(3)(5)
This means that, if we examine the prime factorization of 2x and prime factorization of 2y, they will share exactly ONE 2, ONE 3, and ONE 5.
That is:
2x = (2)(3)(5)(?)(?)(?)
2y = (2)(3)(5)(?)(?)(?)
NOTE: Both prime factorizations might include other primes, BUT
there is no additional overlap beyond the ONE 2, ONE 3, and ONE 5.
Notice that if we divide both sides of both prime factorizations by 2, we get:
x = (3)(5)(?)(?)(?)
y = (3)(5)(?)(?)(?)
Since we already know that
there is no additional overlap beyond the ONE 3, and ONE 5, we can conclude that
the greatest common divisor (GCD) of x and y is 15.
Since we're trying to find the greatest common divisor of x and 2y, we should take a closer look at the prime factorizations of x and 2y:
x = (3)(5)(?)(?)(?)
2y = (2)(3)(5)(?)(?)(?)
We already know that
x and y have
no additional overlap beyond the ONE 3, and ONE 5, the GCD of
x and 2y will be EITHER 15 OR 30
If the prime factorization of x contains a 2, then
x and 2y will share ONE 2, ONE 3, and ONE 5, which means
the GCD of x and 2y will be 30
If the prime factorization of x does not contain a 2, then
x and 2y will share ONE 3, and ONE 5, which means
the GCD of x and 2y will be 15
So, it all comes down to
whether or not the prime factorization of x contains a 2.
Statement 1: y is odd
This information does not tell us
whether or not the prime factorization of x contains a 2
There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = 15 and y = 15. This satisfies the given condition that the GCD of 2x and 2y is 30. In this case
the GCD of x and 2y is 15
Case b: x = 30 and y = 15. This satisfies the given condition that the GCD of 2x and 2y is 30. In this case
the GCD of x and 2y is 30
Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: x is odd
If x is ODD, then we know that
the prime factorization of x does not contain a 2, which means
the GCD of x and 2y is 15
Since we can answer the
target question with certainty, statement 2 is SUFFICIENT
Answer:
B
Cheers,
Brent
