TITLE: $N$ is an at most three digit number whose sum of digits equal to the product of its digits. How many possible values of $N$?
QUESTION [5 upvotes]: The smallest number will be $123$.
The largest number will be $321$.
But how to proceed further, any clue please?

REPLY [6 votes]: For three digits, you rightly determined that $1+2+3 = 6 = 1\cdot2\cdot3$ is the only possible option. You have 6 numbers that you can construct from these digits: 123, 132, 213, 231, 312 and 321.
For two digits, $2+2 = 4 = 2\cdot2$. This yields only one number, 22.
For one digit, all 10 digits yield one solution, so ten in total.
So $N$ has 17 possible values.