If you expand (k+1)^3, you'll see that it's equal to k^3 + 3k^2 + 3k + 1. So,

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

That is, n is one greater than a multiple of k, which means the remainder is 1 when you divide n by k (at least, if k > 1).

In general, I wouldn't have bothered to multiply (k+1)^3 out completely; when you do multiply it out, you may be able to see that every term will have a k in it with the exception of the +1 at the end, and that's all you need to know for this question. So Statement 1) is sufficient, and clearly Statement 2) is not. A.