# Program for Number of solutions to Modular Equations in C/C++?

CC++Server Side ProgrammingProgramming

Here we will see one interesting problem related to modular equations. Suppose we have two values A and B. We have to find number of possible value that the variable X can take, such that (A mod X) = B holds.

Suppose A is 26, and B is 2. So the preferred value for X will be {3, 4, 6, 8, 12, 24} So the count will be 6. That is the answer. Let us see the algorithm to get better idea.

## Algorithm

possibleWayCount(a, b) −

begin
if a = b, then there are infinite solutions
if a < b, then there are no solutions
otherwise div_count := find_div(a, b)
return div_count
end

find_div(a, b) −

begin
n := a – b
div_count := 0
for i in range 1 to square root of n, do
if n mode i is 0, then
if i > b, then
increase div_count by 1
end if
if n / i is not same as i and (n / i) > b, then
increase div_count by 1
end if
end if
done
end

## Example

#include <iostream>
#include <cmath>
using namespace std;
int findDivisors(int A, int B) {
int N = (A - B);
int div_count = 0;
for (int i = 1; i <= sqrt(N); i++) {
if ((N % i) == 0) {
if (i > B)
div_count++;
if ((N / i) != i && (N / i) > B) //ignore if it is already counted
div_count++;
}
}
return div_count;
}
int possibleWayCount(int A, int B) {
if (A == B) //if they are same, there are infinity solutions
return -1;
if (A < B) //if A < B, then there are two possible solutions
return 0;
int div_count = 0;
div_count = findDivisors(A, B);
return div_count;
}
void possibleWay(int A, int B) {
int sol = possibleWayCount(A, B);
if (sol == -1)
cout << "For A: " << A << " and B: " << B << ", X can take infinite values greater than " << A;
else
cout << "For A: " << A << " and B: " << B << ", X can take " << sol << " values";
}
int main() {
int A = 26, B = 2;
possibleWay(A, B);
}

## Output

For A: 26 and B: 2, X can take 6 values