Bunuel wrote:

Is the positive integer x a multiple of 16?

(1) x is a multiple of 8.

(2) x^2 is a multiple of 32.

For a number to be divisible by 16, that number must have at least four 2's in its prime factorisation (because 16 = 2^4).

(1) x is a multiple of 8, or of 2^3. This guarantees that x has at least three 2's, but we cant say about four 2's. Insufficient.

(2) x^2 is a multiple of 32, or of 2^5. Now since x is a positive integer, x^2 is a perfect square, which means all powers of prime numbers in prime factorisation of x^2 will be even. So x^2 cannot just have five 2's, if its a multiple of 32 (2^5), it will have at least six 2's.(x^2 will be at-least 2^6). This in turn tells us that x will be at least 2^3, or we can say that x is a multiple of 8. Same as first statement. Insufficient.

Since both statements tell us the same thing, even after combining them we cant say whether x has more than three 2's or not. So Insufficient.

Hence

E answer