If k is a positive integer, is K the square of an integer?

1) k is divisible by 4

2) k is divisible by exactly four primes



(1)
We know k is a + integer so...
k = 4, 8, 12, 16, 20, 24 ....

INSUFFICIENT since the square root of some of those numbers is an integer (4, 16), but others aren't (8, 12, etc).

(2)
this means that k has four different prime factors, but we don't know how many times those factors appear in the prime factorization of k.
so, for example, if k is 2 x 3 x 5 x 7 (which is divisible by the four primes 2, 3, 5, 7), it's not the square of an integer;
if k is 2 x 2 x 3 x 3 x 5 x 5 x 7 x 7 (divisible by the same four primes), it's the square of the integer 2 x 3 x 5 x 7.
so, insufficient.

(together)
you need the 4 prime factors (because of statement 2), and you also need to have at least two '2's in the prime factorization (because of statement 1).
the aforementioned perfect square (2 x 2 x 3 x 3 x 5 x 5 x 7 x 7) still works.
to create a number that satisfies the criteria yet isn't a perfect square, just add another copy of one of these prime factors (e.g., 2 x 2 x 3 x 3 x 5 x 5 x 7 x 7 x 7).
insufficient

Ans - E