accincognito wrote:
If x and y are integers and 2 < x < y, does y = 16?
(1) The GCF of x and y is 2
(2) The LCM of x and y is 48
Dear
accincognito,
I'm happy to help.
This is a hard problem! See this post for some insight:
http://magoosh.com/gmat/2012/gmat-math-factors/Statement #1:
The GCF of x and y is 2This leave open a wide array of possibilities. All we know is that x and y are two even numbers, both bigger than 2, with no common factors other than two: they could be
x = 4, y = 6
x = 6, y = 8
x = 6, y = 10
x = 6, y = 16
So, it's possible for y to equal 16 or equal something else. This statement, alone and by itself, does not give us sufficient information, so it is
insufficient.
Statement #2:
The LCM of x and y is 48Without any other information, we could have
x = 3, y = 16
x = 4, y = 48
So, it's possible for y to equal 16 or equal something else. This statement, alone and by itself, does not give us sufficient information, so it is
insufficient.
Combined: this is where it gets interesting.
The GCF of x and y is 2The LCM of x and y is 48This is a tricky combination. First, let's list all the factors of 48 --- in order to have a LCM of 48 with another number, each number must be a factor of 48.
factors of 48 = {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}
Those are the possible candidates for x & y. We can eliminate 1 & 2, because x > 2, and we can eliminate 3, because that cannot have a GCF of 2 with anything else.
Possibilities for x & y = {4, 6, 8, 12, 16, 24, 48}
If y = 48, then every other number in the set is factor of 48, so the GCF would be the smaller number --- e.g. the GCF of 6 and 48 is 6. Therefore, we can't use 48.
If y = 24, then the first four numbers are factors of 24, so they don't work, and the GCF of 16 & 24 is 8. Therefore, we can't use 24.
Possibilities for x & y = {4, 6, 8, 12, 16}
Suppose y = 16
x = 4, y = 16 ===> GCF = 4, doesn't work
x = 6, y = 16 ===> GCF = 2 --- this is one possible pair!!
x = 8, y = 16 ===> GCF = 8, doesn't work
x = 12, y = 16 ===> GCF = 4, doesn't work
Suppose y = 8
x = 4, y = 8 ===> GCF = 4, doesn't work
x = 6, y = 8 ===> GCF = 2, but LCM = 24, doesn't work
Suppose y = 6
x = 4, y = 6 ===> GCF = 2, but LCM = 12, doesn't work
So, after all that, the only pair that satisfies both statements is x = 6, y = 16, so it turns out, y does in fact equal 16. We are able to give a definitive answer to the prompt question, so the combined statements are
sufficient.
Answer =
(C)Does all this make sense?
Mike
x,y are positive integers and 2<x<y. Is y=16?