One thing to note here: the two statements say (essentially) the same thing, implying that the answer is either D or E. (This helps a lot when you're forced to guess.)
S1 tells us that x is equal to y times something (let's call that something k), plus 1. Once we have this equation (x = yk + 1), let's unpack it a bit.
We know that yk is divisible by y, so yk is divisible by all of the factors of y.
Now consider yk + 1. If a number is a factor of yk and a factor of (yk + 1), then it must ALSO be a factor of the difference between the two numbers. Here that difference is (yk + 1) - yk = 1, so the number must be a factor of 1. (Let me know if this is still unclear, and I can elaborate on it a bit.)
Since 1 is the only factor of 1, this tells us that yk and yk + 1 share only one factor: 1 itself. Since x = yk + 1, this tells us x and yk share no factors other than 1. Since yk has all the factors of y, including y itself, this tells us that x and y share no factors other than 1. SUFFICIENT!
Statement 2 works in much the same way. (x - y)² = 1 implies that (x - y) = 1 or (x - y) = -1. This gives us x = y + 1 or y = x + 1. In either case, the same logic from S1 applies, and the numbers have only one common factor: 1 itself. SUFFICIENT, and we're done!
