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Find the value of P and modular inverse of Q modulo 998244353

  • Difficulty Level : Hard
  • Last Updated : 27 Sep, 2021

Given two integer P and Q, the task is to find the value of P and modular inverse of Q modulo 998244353. That is 

P * Q^{-1} \% 998244353
 

Note: P and Q are co-prime integers
Examples: 

 

Input: P = 1, Q = 4
Output: 748683265
Explanation:
Refer below for the explanation of the example.

Input: P = 1, Q = 16
Output: 935854081
 

 

 

Approach:  The key observation in the problem is that Q is co-prime with the 998244353, Then Q-1 always exists which can be computed using Extended euclidean Algorithm

 

For Example:
 

 

For P = 1 and Q = 4
We know that,  

 



\frac{1}{4} \equiv 4^{-1} mod 998244353
 

 

 

That is, 4 * 748683265 = 2994733060 equivalent to 1 mod 998244353
Therefore,  1*4^(-1) = 748683265

 

 

Below is the implementation of the above approach:
 

 

C++




// C++ implementation to find the
// value of P.Q -1 mod 998244353
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the value of
// P * Q^-1 mod 998244353
long long calculate(long long p,
                    long long q)
{
    long long mod = 998244353, expo;
    expo = mod - 2;
 
    // Loop to find the value
    // until the expo is not zero
    while (expo) {
 
        // Multiply p with q
        // if expo is odd
        if (expo & 1) {
            p = (p * q) % mod;
        }
        q = (q * q) % mod;
 
        // Reduce the value of
        // expo by 2
        expo >>= 1;
    }
    return p;
}
 
// Driver code
int main()
{
    int p = 1, q = 4;
 
    // Function Call
    cout << calculate(p, q) << '\n';
    return 0;
}

Java




// Java implementation to find the
// value of P.Q -1 mod 998244353
import java.util.*;
 
class GFG{
 
// Function to find the value of
// P * Q^-1 mod 998244353
static long calculate(long p, long q)
{
    long mod = 998244353, expo;
    expo = mod - 2;
 
    // Loop to find the value
    // until the expo is not zero
    while (expo != 0)
    {
 
        // Multiply p with q
        // if expo is odd
        if ((expo & 1) == 1)
        {
            p = (p * q) % mod;
        }
        q = (q * q) % mod;
 
        // Reduce the value of
        // expo by 2
        expo >>= 1;
    }
    return p;
}
 
// Driver code
public static void main(String[] args)
{
    long p = 1, q = 4;
     
    // Function call
    System.out.println(calculate(p, q));
}
}
 
// This code is contributed by offbeat

Python3




# Python3 implementation to find the
# value of P.Q -1 mod 998244353
 
# Function to find the value of
# P * Q^-1 mod 998244353
def calculate(p, q):
     
    mod = 998244353
    expo = 0
    expo = mod - 2
 
    # Loop to find the value
    # until the expo is not zero
    while (expo):
 
        # Multiply p with q
        # if expo is odd
        if (expo & 1):
            p = (p * q) % mod
        q = (q * q) % mod
 
        # Reduce the value of
        # expo by 2
        expo >>= 1
 
    return p
 
# Driver code
if __name__ == '__main__':
     
    p = 1
    q = 4
 
    # Function call
    print(calculate(p, q))
 
# This code is contributed by mohit kumar 29

C#




// C# implementation to find the
// value of P.Q -1 mod 998244353
using System;
class GFG{
  
// Function to find the value of
// P * Q^-1 mod 998244353
static long calculate(long p, long q)
{
    long mod = 998244353, expo;
    expo = mod - 2;
  
    // Loop to find the value
    // until the expo is not zero
    while (expo != 0)
    {
  
        // Multiply p with q
        // if expo is odd
        if ((expo & 1) == 1)
        {
            p = (p * q) % mod;
        }
        q = (q * q) % mod;
  
        // Reduce the value of
        // expo by 2
        expo >>= 1;
    }
    return p;
}
  
// Driver code
public static void Main(string[] args)
{
    long p = 1, q = 4;
      
    // Function call
    Console.WriteLine(calculate(p, q));
}
}
  
// This code is contributed by Ritik Bansal

Javascript




<script>
    // Javascript implementation to find the
    // value of P.Q -1 mod 998244353
     
    // Function to find the value of
    // P * Q^-1 mod 998244353
    function calculate(P, Q)
    {
        let mod = 998244353, expo;
        expo = mod - 2;
        p = 748683265;
 
        // Loop to find the value
        // until the expo is not zero
        while (expo != 0)
        {
 
            // Multiply p with q
            // if expo is odd
            if ((expo & 1) == 1)
            {
                P = (P * Q) % mod;
            }
            Q = (Q * Q) % mod;
 
            // Reduce the value of
            // expo by 2
            expo >>= 1;
        }
        return p;
    }
     
    let p = 1, q = 4;
       
    // Function call
    document.write(calculate(p, q));
 
// This code is contributed by decode2207.
</script>
Output
748683265


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