If z is a multiple of 24, we can think of z as 24 times an integer, or 24i. z^2, then, is (24^2)i. The prime factorization of 24 is (2^3)(3), so the prime factorization of 24^2 is (2^6)(3^2). 3^2 is 9, so 24 squared is divisible by 9. It follows that any multiple of 24 squared is divisible by 9 as well.

So, since z^2 is divisible by 9, the remainder when it is divided by 9 is 0, choice (A).