Full factoring

From the answer choices, there is only one logical answer. 3600 is a perfect square, and only perfect squares have an odd number od factors. This narrows it down to 43 or 45. When breaking down 3600 into prime factorization, each factors exponent gets one added to it, and then those numbers are multiplied together. Thus, the number must be composite, so 45 is the solution.

$3600=2^4 \times 3^2 \times 5^2$ . Thus number of factors is $5 \times 3 \times 3=\boxed{45}$ .