求P的值和Q模998244353的模逆

给定两个整数P和Q，任务是找到P的值和Q模998244353的Q的模逆。

$P * Q^{-1} \% 998244353$

注意： P和Q是互素整数
例子：

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

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

方法：问题中的关键发现是Q与998244353互质，那么Q ^-1始终存在，可以使用扩展欧几里德算法进行计算

例如：

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

下面是上述方法的实现：

C++

// C++ implementation to find the
// value of P.Q -1 mod 998244353
 
#include 
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

输出

748683265