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

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

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,  

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

Time Complexity: O(log(expo)), where the expo is calculated in the program
Auxiliary Space: O(1), as no extra space is used