imc

Since we want $n$ to be as large as possible, we would like $d$ in $10d+e$ to be as large as possible. So, $d=7,$ the greatest single-digit prime. Then, $e$ cannot be $5$ because $10(7)+5=75,$ which is not prime. So $e=3$. Therefore, $d\cdot e\cdot (10d+e)=7\cdot 3\cdot 73=1533$. So, the sum of the digits of $n$ is $1+5+3+3=12\Rightarrow \boxed{12}$ ~ MathGenius_ (Edited by Sophia866)