IMO : C

Statement 1 :
K is divisible by 4
K can be 4,8,12,16 etc,may or may not be a perfect square.
Insufficient.

Statement 2 :
K is divisible by exactly 4 different primes
picking Numbers
210 is divisible by 2,3,5,7 but not a perfect square
44100 is divisible by 2,3,5,7 and is a perfect square.
Hence Insufficient.

Combining 1 and 2

If the number is dicisible by 4 different primes and is divisible by 4 it will be perfect square.

Eg : 44100
or
13*13*2*2*5*5*7*7 is divisible by 4 and a perfect square.
Similarly other numbers.

Whats the OA?

Not sure about the OA because it was on a GMAC test which I have reset. The word "exactly" confuses me. So combining 1 and 2 can't K be 2*2*3*5*7=420 which is divisible by 4 and 4 different primes but not a perfect square??

IMO E.

1&2

2*2*3*5*7 meets both statements, but is not a perfect square. 2*2*3*5*7*3*5*7 is a perfect square.

Guys can you explain why are you using prime number more than once?
statement 2 says K is divisible by EXACTLY 4 different prime numbers
shouldn't that mean, there are only 4 distinct prime numbers?

In order to be divisible by 4 different primes and the number 4, we would at least have to square 2. I think the answer is E because if we square 2 only then No to the question (which implies that we can square the other terms); if we square all the primes then yes to the question.