Quotient by Digit Sum

Number Theory Level 3

Let \(S(N) \) denote the digit sum of integer \(N\). Let \(M\) denote the maximum value of \( \frac {N} {S(N)} \), where \(N\) is a 3-digit number. How many 3-digit numbers \(N\) satisfy \( \frac {N}{S(N)} = M\)?

Details and assumptions

The digit sum of an integer is the sum of all its digits. For example, the digit sum of \(N = 1123\) is \(1+1+2+3=7\).

The number \(12=012\) is a 2-digit number, not a 3-digit number.


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