Find First element in AP which is multiple of given Prime in C++

Concept

With respect of given first term (A) and common difference (d) of an Arithmetic Progression, and a prime number (P), our task is to determine the position of the first element in the given AP which is treated as a multiple of the given prime number P.

Input

A = 3, d = 4, P = 5

Output

3

Explanation

The fourth term of the given AP is a multiple of prime number 5.

First Term = 3

Second Term = 3+4 = 7

Third Term = 3+2*4 = 11

Fourth Term = 3+3*4 = 15

Method

Assume the term be AN. As a result of this,

AN = (A + (N-1)*d)

So, it is given that AN is a multiple of P. As aresult of this,

A + (N-1)*d = l*P

Here, l is a constant.

So assume A be (A % P) and d be (d % P). Now, we have (N-1)*d = (l*P – A).

With the help of adding and subtracting P on RHS, we obtain −

(N-1)*d = P(l-1) + (P-A),

In this case, P-A is treated as a non-negative number

(because A is replaced by A%P which is smaller than P) At last taking mod on both sides −

((N-1)*d)%P = (P-A)%P or, ((N-1)d)%P = P-A

Assume calculate a Y < P, so that (d*Y)%P = 1. Now, this Y is termed as the inverse modulo of d with respect to P.

Finally answer N is −

((Y*(P-A)) % P) + 1.

Example

 Live Demo

#include <bits/stdc++.h>
using namespace std;
// Shows iterative Function to calculate
// (x1^y1)%p1 in O(log y1) */
int power(int x1, int y1, int p1){
   // Used to initialize result
   int res1 = 1;
   // Used to update x if it is more than or
   // equal to p
   x1 = x1 % p1;
   while (y1 > 0) {
      // It has been seen that if y1 is odd, multiply x1 with
      result
      if (y1 & 1)
         res1 = (res1 * x1) % p1;
         // y1 must be even now
      y1 = y1 >> 1; // y1 = y1/2
      x1 = (x1 * x1) % p1;
   }
   return res1;
}
// Shows function to find nearest element in common
int NearestElement1(int A, int d, int P){
   // Shows base conditions
   if (A == 0)
      return 0;
   else if (d == 0)
      return -1;
   else {
      int Y = power(d, P - 2, P);
      return (Y * (P - A)) % P;
   }
}
// Driver code
int main(){
   int A = 3, d = 4, P = 5;
   // Used to module both A and d
   A %= P;
   d %= P;
   // Shows function call
   cout << NearestElement1(A, d, P);
   return 0;
}

Output

3