smc

Lets begin by noticing that: $$\frac{3p+25}{2p-5}=1+\frac{p+30}{2p-5}$$ Therefore, in order for $\frac{3p+25}{2p-5}$ to be a positive integer, $\frac{p+30}{2p-5}$ must be a non-negative integer. Since the bottom the the fraction is an odd number, we can multiply the top of $\frac{p+30}{2p-5}$ by 2 without changing whether it is an integer or not. Therefore, in order for $\frac{3p+25}{2p-5}$ to be an integer, $\frac{2p+60}{2p-5}=1+\frac{65}{2p-5}$ must also be an integer. As a result, $2p-5$ must be a factor of 65, or $p=3,5,9,35$. Therefore $p$ must be at least 3, and less than or equal to 35. So the answer which best fits these constraints is $\boxed{\text{at least }3\text{ and no more than }35}$.