Data Sufficiency

If you know remainder arithmetic ("modular arithmetic"), this is a three-second question, because if you're dividing by k, then "k+1" is the same as "1", so (k+1)^3 is the same as 1^3, and our remainder is 1 using Statement 1.

But that won't make any sense to most test takers, so we can also expand the right side of Statement 1:

(k+1)^3 = k^3 + 3k^2 + 3k + 1

Notice that everything here is a multiple of k besides the "+1" at the end. So n is exactly 1 larger than some multiple of k, which is exactly what we mean when we say "the remainder is 1 when we divide n by k". So Statement 1 is sufficient. Statement 2 isn't useful, so the answer is A.