Count of numbers between range having only non-zero digits whose sum of digits is N and number is divisible by M
Given a range [L, R] and two positive integers N and M. The task is to count the numbers in the range containing only non-zero digits whose sum of digits is equal to N and the number is divisible by M.
Input: L = 1, R = 100, N = 8, M = 2
Only 8, 26, 44 and 62 are valid numbers
Input: L = 1, R = 200, N = 4, M = 11
Only 22 and 121 are valid numbers
Prerequisites : Digit DP
Approach: Firstly, if we are able to count the required numbers up to R i.e. in the range [0, R], we can easily reach our answer in the range [L, R] by solving for from zero to R and then subtracting the answer we get after solving for from zero to L – 1. Now, we need to define the DP states.
- Since we can consider our number as a sequence of digits, one state is the position at which we are currently in. This position can have values from 0 to 18 if we are dealing with numbers up to 1018. In each recursive call, we try to build the sequence from left to right by placing a digit from 0 to 9.
- Second state is the sum of the digits we have placed so far.
- Third state is the remainder which defines the modulus of the number we have made so far modulo M.
- Another state is the boolean variable tight which tells the number we are trying to build has already become smaller than R so that in the upcoming recursive calls we can place any digit from 0 to 9. If the number has not become smaller, the maximum limit of digit we can place is digit at the current position in R.
For the number to have only non-zero digits, we maintain a variable nonz whose value if 1 tells the first digit in the number we have placed is a non-zero digit and thus, now we can’t place any zero digit in upcoming calls. Otherwise, we can place a zero digit as a leading zero so as to make number of digits in current number smaller than number of digits in upper limit.
Below is the implementation of the above approach:
Time Complexity: O(M*M)
Auxiliary Space: O(M*M)
Short Python Implementation :