imc

Let the original price be $x$. Then, for option $1$, the discounted price is $(1-.15)(1-.15)x=.7225x$. For option $2$, the discounted price is $(1-.1)(1-.1)(1-.1)x=.729x$. Finally, for option $3$, the discounted price is $(1-.25)(1-.05)=.7125x$. Therefore, $n$ must be greater than $\max (x-.7225x,x-.729x,x-.7125x)$. It follows $n/100$ must be greater than $.2875$. We multiply this by $100$ to get the percent value, and then round up because $n$ is the smallest integer that provides a greater discount than $28.75$, leaving us with the answer of $\boxed{29}$