Hi alanforde800Maximus,
This DS question can be solved by a combination of TESTing VALUES and doing a bit of Algebra. We're told that X and Y are POSITIVE INTEGERS. We're asked for the Greatest Common Factor (GCF) of X and Y.
1) When X is divided by Y, the remainder is 1.
This Fact tells us that X = Y +1. This piece of information involves a relatively rare math concept - and one that you probably will not see on the GMAT: when dealing with two consecutive, positive integers, the GCF will always be 1 (meaning that NO other integers will divide evenly into both numbers). Even if you don't know that rule though, it's easy enough to prove. Here are some examples:
X=4, Y=3... X is divisible by 1, 2 and 4... Y is divisible by 1 and 3... the GCF is 1
X=5, Y=4... X is divisible by 1 and 5... Y is divisible by 1, 2 and 4... the GCF is 1
X=6, Y=5... X is divisible by 1, 2, 3 and 6... Y is divisible by 1 and 5... the GCF is 1
Etc.
Thus, the answer to the question is ALWAYS 1.
Fact 1 is SUFFICIENT
2) X^2 - 2XY + Y^2 = 1
This quadratic equation can be 'reverse-FOILed' into...
(X - Y)(X - Y) = 1
(X - Y)^2 = 1
(X - Y) = 1 or -1
Since X and Y differ by 1 or -1, and they're both positive integers, we have the same situation that we have in Fact 1 (above).
Fact 2 is SUFFICIENT
Final Answer: D
GMAT assassins aren't born, they're made,
Rich
