Problem:
Problem F: Find the Multiples

You are given a sequence a0a1...aN-1 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.

Input

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.

Output

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

Sample Input

3 32 64 7
4 35 89 5
5 555 442 3
5 777 465 11
100000 666 701622763 65537
0 0 0 0

Output for the Sample Input

2
4
6
3
68530

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.

Solution in Ruby (judge verdict: Accepted)

class Array
	def accum
		ac = Array.new(size + 1)
		ac[0] = 0
		(1..size).each do |i|
			ac[i] = ac[i-1] + self[i-1]
		end
		ac
	end
end

loop do
	n, s, w, q = gets.split.map(&:to_i)
	break if n == 0

	g = s
	a = Array.new(n)
	(0..n-1).each do |i|
		a[i] = g / 7 % 10
		g = (g.even? ?  g / 2 : (g / 2) ^ w)
	end
	a.reverse!

	k = 0
	if q == 2 || q == 5
		t = 0
		a.each_with_index do |x, i|
			if x % q == 0
				k += n - i 
				t += 1
				k -= t if x == 0
			end
		end
		puts k
		next
	end

	t = 1
	mod = a.map do |x|
		y = x * t % q
		t = t * 10 % q
		y
	end
	mod_accum = mod.accum.map {|x| x % q}

	count = Hash.new(0)
	(0..n).each do |i|
		x = mod_accum[i]
		k += count[x] if i > 0 && a[i-1] > 0
		count[x] += 1
	end
	p k
end