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Frobenius coin problem
  • Difficulty Level : Hard
  • Last Updated : 04 Feb, 2019

Given two coins of denominations “X” and “Y” respectively, find the largest amount that cannot be obtained using these two coins (assuming infinite supply of coins) followed by the total number of such non obtainable amounts, if no such value exists print “NA”.

Examples :

Input : X=2, Y=5  
Output: Largest amount = 3
        Total count  = 2
We cannot represent 1 and 3 from infinite supply
of given two coins. The largest among these 2 is 3.
We can represent all other amounts for example 13
can be represented 2*4 + 5.

Input : X=5, Y=10
Output: NA
There are infinite number of amounts that cannot
be represented by these two coins.

We strongly recommend that you click here and practice it, before moving on to the solution.

One important observation is, if GCD of X and Y is not one, then all values that can be formed by given two coins are multiples of GCD. For example if X = 4 and Y = 6. Then all values are multiple of 2. So all values that are not multiple of 2, cannot be formed by X and Y. Thus there exist infinitely many values that cannot be formed by 4 and 6, and our answer becomes “NA”.

This general problem for n coins is known as classic Forbenius coin problem.

When the number of coins is two, there is 
explicit formula if GCD is not 1. The formula
is:
  Largest amount A = (X * Y) - (X + Y)
  Total amount = (X -1) * (Y - 1) /2 
 

Hence, we can now easily answer the above question by following the below steps:



  1. Calculate GCD of X and Y
  2. If GCD is 1 then required largest amount is (X*Y)-(X+Y) and total count is (X-1)*(Y-1)/2
  3. Else print “NA”

    Below is a the program based on the same.

    C++




    // C++ program to find the largest number that
    // cannot be formed from given two coins
    #include <bits/stdc++.h>
    using namespace std;
      
    // Utility function to find gcd
    int gcd(int a, int b)
    {
        int c;
        while (a != 0)
        {
            c = a;
            a = b%a;
            b = c;
        }
        return b;
    }
      
    // Function to print the desired output
    void forbenius(int X,int Y)
    {
        // Solution doesn't exist 
        // if GCD is not 1
        if (gcd(X,Y) != 1)
        {
            cout << "NA\n";
            return;
        }
      
        // Else apply the formula
        int A = (X*Y)-(X+Y);
        int N = (X-1)*(Y-1)/2;
      
        cout << "Largest Amount = " << A << endl;
        cout << "Total Count = " << N << endl;
    }
      
    // Driver Code
    int main()
    {
        int X = 2,Y = 5;
        forbenius(X,Y);
      
        X = 5, Y = 10;
        cout << endl;
        forbenius(X,Y);
        return 0;
    }

    Java




    // Java program to find the largest
    // number that cannot be formed 
    // from given two coins
    import java.io.*;
      
    class GFG
    {
    // Utility function to find gcd
        static int gcd(int a, int b)
        {
            int c;
            while (a != 0)
            {
                c = a;
                a = b % a;
                b = c;
            }
            return b;
        }
      
        // Function to print the
        // desired output
        static void forbenius(int X, 
                              int Y)
        {
            // Solution doesn't exist 
            // if GCD is not 1
            if (gcd(X, Y) != 1)
            {
                System.out.println( "NA");
                return;
            }
          
            // Else apply the formula
            int A = (X * Y) - (X + Y);
            int N = (X - 1) * (Y - 1) / 2;
          
            System.out.println("Largest Amount = " + A );
            System.out.println("Total Count = " + N );
        }
          
        // Driver Code
        public static void main(String[] args)
        {
            int X = 2,Y = 5;
            forbenius(X, Y);
            X = 5;
            Y = 10;
            System.out.println();
            forbenius(X, Y);
              
        }
    }
      
    // This code is contributed by Sam007

    Python3




    # Python3 program to find the largest 
    # number that cannot be formed
    # from given two coins
      
    # Utility function to find gcd
    def gcd(a, b):
        while (a != 0):
            c = a;
            a = b % a;
            b = c;
          
        return b;
      
    # Function to print the desired output
    def forbenius(X, Y):
      
        # Solution doesn't exist 
        # if GCD is not 1
        if (gcd(X, Y) != 1):
            print("NA");
            return;
      
        # Else apply the formula
        A = (X * Y) - (X + Y);
        N = (X - 1) * (Y - 1) // 2;
      
        print("Largest Amount =", A);
        print("Total Count =", N);
      
    # Driver Code
    X = 2
    Y = 5;
    forbenius(X, Y);
      
    X = 5
    Y = 10;
    print("");
    forbenius(X, Y);
      
    # This code is contributed by mits

    C#




    // C# program to find the largest
    //  number that cannot be formed 
    // from given two coins
    using System;
      
    class GFG
    {
    // Utility function to find gcd
        static int gcd(int a, int b)
        {
            int c;
            while (a != 0)
            {
                c = a;
                a = b%a;
                b = c;
            }
            return b;
        }
      
        // Function to print the
        // desired output
        static void forbenius(int X, int Y)
        {
            // Solution doesn't exist 
            // if GCD is not 1
            if (gcd(X,Y) != 1)
            {
                Console.WriteLine( "NA");
                return;
            }
          
            // Else apply the formula
            int A = (X * Y) - (X + Y);
            int N = (X - 1) * (Y - 1) / 2;
          
            Console.WriteLine("Largest Amount = " + A );
            Console.WriteLine("Total Count = " + N );
        }
          
          
          
      
        // Driver Code
        public static void Main()
        {
            int X = 2,Y = 5;
            forbenius(X,Y);
            X = 5;
            Y = 10;
            Console.WriteLine();
            forbenius(X,Y);
          
        }
    }
          
    // This code is contributed by Sam007

    PHP




    <?php
    // php program to find the largest 
    // number that cannot be formed
    // from given two coins
      
    // Utility function to find gcd
    function gcd($a, $b)
    {
        $c;
        while ($a != 0)
        {
            $c = $a;
            $a = $b % $a;
            $b = $c;
        }
          
        return $b;
    }
      
    // Function to print the desired output
    function forbenius($X, $Y)
    {
        // Solution doesn't exist 
        // if GCD is not 1
        if (gcd($X, $Y) != 1)
        {
            echo "NA\n";
            return;
        }
      
        // Else apply the formula
        $A = ($X * $Y) - ($X + $Y);
        $N = ($X - 1) * ($Y - 1) / 2;
      
        echo "Largest Amount = ", $A, "\n";
        echo "Total Count = ", $N, "\n";
    }
      
    // Driver Code
      
        $X = 2; $Y = 5;
        forbenius($X, $Y);
      
        $X = 5; $Y = 10;
        echo "\n";
        forbenius($X, $Y);
      
    // This code is contributed by ajit.
    ?>


    Output :
    Largest Amount = 3
    Total Count = 2
    
    NA
    

    References:
    https://en.wikipedia.org/wiki/Coin_problem

    This article is contributed by Ashutosh Kumar. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

    Attention reader! Don’t stop learning now. Get hold of all the important mathematical concepts for competitive programming with the Essential Maths for CP Course at a student-friendly price. To complete your preparation from learning a language to DS Algo and many more,  please refer Complete Interview Preparation Course.

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