Minimum Digit Sum Fraction

Algebra Level 3

Let $S(N)$ denote the digit sum of the integer $N$ . As $N$ ranges over all 3-digit positive numbers, what value of $N$ would give the minimum of $M = \frac{N}{S(N)}$ ?

Details and assumptions

The number $12=012$ is a 2-digit number, not a 3-digit number.

The digit sum of a number is the sum of all its digits. For example the digit sum of 1123 is $1 + 1 + 2 + 3 = 7$ .