Find the Multiples

You are given a sequence a0a1···aN−1 of digits and a prime number Q. For each i ≤ j with ai ≠ 0, the subsequence aiai+1···aj can be read as a decimal representation of a positive integer. Subsequences with leading zeros are not considered. Your task is to count the number of pairs (i, j) such that the corresponding subsequence is a multiple of Q.

The input consists of at most 50 datasets. Each dataset is represented by a line containing four integers N, S, W, and Q, separated by spaces, where 1 ≤ N ≤ 105, 1 ≤ S ≤ 109, 1 ≤ W ≤ 109, and Q is a prime number less than 108. The sequence a0 ··· aN−1 of length N is generated by the following code, in which ai is written as a[i].

int g = S;for(int i=0; i<N; i++) {    a[i] = (g/7) % 10;    if( g%2 == 0 ) { g = (g/2); }    else           { g = (g/2) ^ W; }}

Note: the operators /, %, and ^ are the integer division, the modulo, and the bitwise exclusiveor, respectively. The above code is meant to be a random number generator. The intended solution does not rely on the way how the sequence is generated.

The end of the input is indicated by a line containing four zeros separated by spaces.

For each dataset, output the answer in a line. You may assume that the answer is less than 230.

In the first dataset, the sequence is 421. We can find two multiples of Q = 7, namely, 42 and 21.

In the second dataset, the sequence is 5052, from which we can find 5, 50, 505, and 5 being the multiples of Q = 5. Notice that we don’t count 0 or 05 since they are not a valid representation of positive integers. Also notice that we count 5 twice, because it occurs twice in different positions.

In the third and fourth datasets, the sequences are 95073 and 12221, respectively.