Is positive integer k the square of an integer?

You're absolutely right; it's not out of scope.

When I first created the question, statement 1 read "k = n!, where n is an integer greater than 1"
Then, as I was posting the question, I realized that making the number prime would require fewer words. In doing so, I made the question MUCH easier.
My bad.

Cheers,
Brent

IF YOU FIND MY SOLUTION HELPFUL PLEASE GIVE ME KUDOS

(1) k = p! where p is a prime number. This means that p = 2,3,5,7.... let p =2 k =2, so we get a no. Let p=3, then k = 3*2 = 6 so we get another no. Let p =5, then k =5*4*3*2*1 = 1*120, let p = 7. Then k = 7*6*5*4*3*2*1. The pattern that the test is trying to get you to see here, is that no matter what prime we pick, we will have a single leading prime factor, thus K can never be the square of an integer sufficient

(2) k+1 = w^2-2w+2 = (w-1)^2. Notice that there are no restrictions on w, or (w-1)^2 directly. Our only restriction is that k must be a positive integer.Then let (w-1)^2 = 0, (obtained when w=1)

This is an upper level 700 question for sure, but not nearly as bad as some of the 3 overlapping set problems. In my opinion, it is not unrealistic for an upper level test taker to run into a problem like this on test day.