If we prime factorize everything:lrojas wrote:If the positive integer x divisible by 200?
(1) x^2 is divisible by 5000
(2) x^3 is divisible by 640
The question asks if x is divisible by 2^3 * 5^2
From S1: x^2 is divisible by 2^3 * 5^4. For x^2 to be divisible by 2^3, it will need to be true that x itself is divisible by at least 2^2, since if x were only divisible by 2^1, then x^2 would only be divisible by 2^2. So if x^2 is divisible by 2^3 * 5^4, that guarantees that x is divisible at least by 2^2 * 5^2. That's not quite sufficient.
From S2: x^3 is divisible by 2^7 * 5^1. As above, if this is true, x cannot only be divisible by 2^2, because then x^3 would only be divisible by 2^6. It must be true that x is divisible at least by 2^3, and also by 5, so this statement guarantees x is divisible at least by 2^3 * 5. Again this is not quite sufficient.
Combining the information from both statements, if x is divisible both by 2^2 * 5^2 and by 2^3 * 5, it must be divisible by the LCM of those two numbers, so must be divisible by 2^3 * 5^2, which is what we wanted to know. So the answer is C.
