Skip to content
Related Articles

Related Articles

Frobenius coin problem

View Discussion
Improve Article
Save Article
  • Difficulty Level : Medium
  • Last Updated : 03 Aug, 2022
View Discussion
Improve Article
Save Article

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 count = (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 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.
?>

Javascript




// JavaScript program to find the largest number that
// cannot be formed from given two coins
 
// Utility function to find gcd
function gcd(a, b)
{
    let 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)
    {
        console.log("NA");
        return;
    }
   
    // Else apply the formula
    let A = (X*Y)-(X+Y);
    let N = (X-1)*(Y-1)/2;
   
    console.log("Largest Amount = " + A);
    console.log("Total Count = " + N);
}
   
// Driver Code
let X = 2,Y = 5;
forbenius(X,Y);
 
console.log();
X = 5, Y = 10;
forbenius(X,Y);
 
// This code is contributed by phasing17

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


My Personal Notes arrow_drop_up
Recommended Articles
Page :

Start Your Coding Journey Now!