smc

Product $P$ can be written as $2n-1$,$2n+1$,$2n+3$. Because $P$ is defined as a "3 consecutive odd integer" product impies that $P$ must be divisible by at least 3. A is ruled out because factors of 5 only arise every 5 terms, if we were to take the 3 terms in the middle of the factors of 5 we wouldn't have a factor of 5. Obviously B is impossible because we are multiplying odd numbers and 2 would never become one of our prime factors. C is ruled out with the same logic as A. Lastly E is ruled out because we have already proved that 3 is possible, and the question asks for greatest possible. $\boxed{3}$ * In first step we could also apply modular arthmetic and get the same answer.