If you chose E, you chose the right answer. If the source thinks the answer is B, the source is simply wrong.
Statement 1 means that x^2 has an odd number of divisors. If you look at a number like 9, say, it has three divisors in total: 1, 3 and 9. So 9 is a number which has an odd number of divisors. A number like 15, on the other hand, has the divisors 1, 3, 5 and 15. So 15 has four divisors, and so has an even number of divisors.
Every perfect square has an odd number of divisors, so Statement 1 tells us nothing useful at all here. Statement 2 is also not sufficient: it might be that x=5, and x is prime, or it might be that x=15 and x is not prime. So the answer is E.
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