Implement the Pseudo-codes of Euclid’s Algorithm

int exeuclid(int a, int m)
{
     int A3 = m;
     int A2 = 0, A1 = 1;

     if (m == 1)
          return 0;

     while (a &gt; 1)
     {
          // q is quotient
          int q = a / m;

          int t = m;

          // m is remainder now, process
          // same as Euclid's algo
          m = a % m;
          a = t;
          t = A2;

          // Update A1 and A2
          A2 = A1 - q * A2;
          A1 = t;
      }

     // Make A1 positive
     if (A1 &lt; 0)
     A1 += A3;
     return A1;
}

Let us look into how to implement Euclid’s algorithm and Euclid’s Extended Algorithms using Java Programming Language.

Euclid’s Algorithm Pseudocode

int euclid(int a, int b){
     if(b==0)
          return a;
     else
          return euclid(b, a%b);
}

Euclid’s Extended Algorithm Pseudocode

int exeuclid(int a, int m)
{
    int A3 = m;
    int A2 = 0, A1 = 1;

    if (m == 1)
        return 0;

    while (a &gt; 1)
    {
        // q is quotient
        int q = a / m;

        int t = m;

        // m is remainder now, process
        // same as Euclid's algo
        m = a % m;
        a = t;
        t = A2;

        // Update A1 and A2
        A2 = A1 - q * A2;
        A1 = t;
    }

    // Make A1 positive
    if (A1 &lt; 0)
        A1 += A3;
    return A1;
}

Question

Implement the pseudo-codes of Euclid’s algorithm with recursive function and extended Euclid’s algorithm in any programming language you are comfortable with. Your program should take two integers A and B as inputs and give either as GCD (A, B) = d or B-1 = y.

1) Determine the

a) gcd(72345, 43215)

b) gcd(2740, 1760)

2) Find the multiplicative inverse of

a. 550 mod 1769

b. 7465 mod 2464

c. 42828 mod 6407

Sample Program

import java.util.Scanner;
class Lab3{
     //euclids algorithm
     int euclid(int a, int b){
          if(b==0)
               return a;
          else
               return euclid(b, a%b);
          }
     //euclids extended algorithm
     int exeuclid(int a, int m)
     {
          int A3 = m;
          int A2 = 0, A1 = 1;
          if (m == 1)
               return 0;

          while (a &gt; 1)
          {
               // q is quotient
               int q = a / m;

               int t = m;

               // m is remainder now, process
               // same as Euclid's algo
               m = a % m;
               a = t;
               t = A2;

               // Update A1 and A2
               A2 = A1 - q * A2;
               A1 = t;
          }

          // Make A1 positive
          if (A1 &lt; 0)
          A1 += A3;
          return A1;
     }
     public static void main(String[] args) {
          Lab3 ob = new Lab3();
          int x, y, choice;

          Scanner input = new Scanner(System.in);
          System.out.println("Enter Choice \n Enter 1 for gcd and 2 for extended ed");
          choice=input.nextInt();
          switch(choice)
          {
               case 1:
                    int output;
                    System.out.println("Enter the first number");
                    x = input.nextInt();
                    System.out.println("Enter the second number");
                    y = input.nextInt();
                    output = ob.euclid(x, y);
                    System.out.println("GCD is:"+output);
                    break;
               case 2:
                    System.out.println("To Find Inverse");
                    int m, b;
                    System.out.println("Enter m");
                    m=input.nextInt();
                    System.out.println("Enter b");
                    b = input.nextInt();
                    System.out.println("Inverse is equal to: "+ob.exeuclid(m, b));
                    break;
               default:
                    System.out.println("Wrong Choice");
          }

     }
}

Sample Output

Implementing Euclid's Algorithm — Implementing Euclid’s Algorithm

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