To solve this quickly, we must realize that the greatest common factor of two consecutive integers must be 1. The reason is simple: if X and Y have a common factor k, then they can be expressed as k*a and k*b, and the difference X-Y = k(a-b). Notice that this difference, k(a-b), is a multiple of k. In other words, the difference between X and Y must be a multiple of all the common factors of X and Y. For instance 12 and 18 have common factors 1, 2, 3 and 6. The difference between 12 and 18 is a multiple of all these factors. This means that if the difference between X and Y is 1, the only possible common factor is 1.

Ok, back to the question

Statement 1:
x = y*q + 1. the greatest common factor of x and y*q is 1. Since x and yq don't have any shared factor other than 1, x and y don't either. The greatest common factor of x and y must be 1.
SUFFICIENT

Statement 2:
x^2 - 2xy + y^2 = (x-y)^2 = 1. This statement tells us that the square of (x-y) is 1, so x-y = 1 or -1. The positive difference between x and y is 1, so the greatest common factor must be 1.
SUFFICIENT

The answer is D

To find: GCD of x & y

Statement 1:
x/y = remainder 1
that means that is no common factor greater than 1
SUFFICIENT

Statement 2:
x^2-2xy+y^2=1
(x - y)^2 = 1
Either x - y is 1 or x - y = -1
this implies that x & y are consecutive Ints
that means these is no common factor except 1.
SUFFICIENT

[spoiler]{D}[/spoiler]

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!