Dear buoyant,buoyant wrote:Is x^2-y^2 = EVEN ?
1) x+y = ODD
2) x-y = ODD
I'm happy to respond.
What's curious about this question is that we get no statement about the nature of x & y. It is usually a lethal mistake on the GMAT to assume, for example, that x & y are integers when that is not specified. Suppose they are pure numbers --- maybe positive or negative or zero, maybe integers or fractions or decimals. In that case, there a continuous infinity of decimals that are neither even nor odd.
For Statement #1
Try x = 1 and y = 0. Then x + y = 1, which is ODD, and x^2-y^2 = 1, which is ODD.
BUT, we could pick
x = y = 1.5. Then x + y = 1, which is ODD, and x^2-y^2 = 0, which is EVEN.
Different choices lead to different answers. This statement, by itself, is insufficient.
For Statement #2
Try x = 1 and y = 0. Then x - y = 1, which is ODD, and x^2-y^2 = 1, which is ODD.
With two integers, we have to have one even and one odd; the squares will remain even and odd, respectively, and the difference will always be ODD.
For non-integer values:
Try x = 2.5 and y = 1.5. Then x - y = 1, which is ODD, and x^2-y^2 = 6.25 - 2.25 = 4, which is EVEN.
Different choices lead to different answers. This statement, by itself, is insufficient.
For combined statements.
As above, a pair of integers, one even and one odd, will satisfy with statements and always produce an odd difference of squares --- recall that (x - y)*(x + y) = x^2-y^2. See:
https://magoosh.com/gmat/2012/gmat-quant ... o-squares/
https://magoosh.com/gmat/2013/three-alge ... -the-gmat/
Curiously, it is impossible to produce a pair of non-integer values that simultaneously have an ODD sum and and ODD difference. In other words, it's impossible to find a non-integer pair that simultaneously satisfies both statements. If both statements are true, then x & y must be integers, and the difference of squares is ODD. Thus, we would have a definitive answer to the prompt.
Together, the statements are sufficient. Answer = [spoiler](C)[/spoiler]
Does all this make sense?
Mike
