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

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.

9. The maximum value of N/S(N) is 100, and for N = 100, 200, 300, 400, ..., 900, we get the desired value of N/S(N).

To find the number of 3-digit numbers N that satisfy N/S(N) = M, we can break down the problem into steps:

Step 1: Generate all possible 3-digit numbers.

Since N is a 3-digit number, it can range from 100 to 999 inclusive.

Step 2: Calculate S(N) for each generated number.

To calculate S(N), we need to find the sum of all the digits in N. We can do this by taking the modulus of 10 repeatedly and adding the remainders until N becomes zero.

For example, let's calculate S(N) for N = 1123:
- N % 10 = 3 (last digit), add 3 to the sum
- N = N / 10 = 112 (removing the last digit)
- N % 10 = 2, add 2 to the sum
- N = N / 10 = 11
- N % 10 = 1, add 1 to the sum
- N = N / 10 = 1
- N % 10 = 1, add 1 to the sum
- N = N / 10 = 0 (N is now zero)
The sum of the digits is 7.

Step 3: Calculate N/S(N) for each generated number.

Using the calculated S(N) from step 2, we can calculate N/S(N) for each generated number N.

Step 4: Find the maximum value M of N/S(N) among all generated numbers.

By comparing the calculated N/S(N) values, we can find the maximum value M.

Step 5: Count the numbers that satisfy N/S(N) = M.

Finally, count the number of 3-digit numbers N that satisfy N/S(N) = M based on the calculated values in step 4.

Applying these steps systematically will give you the desired number of 3-digit numbers N that satisfy N/S(N) = M.