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Sum of minimum value of x and y satisfying the equation ax + by = c
  • Difficulty Level : Hard
  • Last Updated : 19 Apr, 2021

Given three integers a, b, and c representing a linear equation of the form ax + by = c, the task is to find the solution (x, y) of the given equation such that (x + y) is minimized. If no solution exists for the above equation then print “-1”
Note: x and y are positive integers.

Examples: 

Input: a = 2, b = 2, c = 0 
Output:
Explanation: 
The given equation is 2x + 2y = 0. 
Therefore, x = 0 and y = 0 is the required solution with minimum value of (x + y).

Input: a = 2, b = 2, c = 1 
Output: -1 
Explanation: 
The given equation is 2x + 2y = 1. 
No solution exists for the given equation for positive values of x and y. 
 

Approach: To solve the above problem, find any solution say (x, y) of the given Linear Diophantine Equation and then accordingly find value of x and to minimised the sum.
Below is the solution (x’, y’) of the given equation: 



x' = x + k * \frac{b}{g}          [Tex]y’ = y – k * \frac{a}{g}          [/Tex]
 

where g is gcd(a, b) and k is any integer. 

x' + y' = x + y + k*(\frac{b}{g} - \frac{a}{g})          [Tex]x’ + y’ = x + y + k*\frac{b-a}{g}         [/Tex]

From the above equation we observe that: 

  1. If a is less than b, we need to select the smallest possible value of K.
  2. Else, if a is greater than b, we need to select the largest possible value of K.
  3. If a = b, all solutions will have the same sum (x + y).

Below is the implementation of the above approach:

C++




// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the gcd of a & b
// by Euclid's method
 
// x and y store solution of
// equation ax + by = g
int gcd(int a, int b, int* x, int* y)
{
 
    if (b == 0) {
        *x = 1;
        *y = 0;
        return a;
    }
    int x1, y1;
    int store_gcd = gcd(b, a % b,
                        &x1, &y1);
 
    // Euclidean Algorithm
    *x = y1;
    *y = x1 - y1 * (a / b);
 
    // store_gcd returns
    // the gcd of a and b
    return store_gcd;
}
 
// Function to find any
// possible solution
int possible_solution(int a, int b,
                    int c, int* x0,
                    int* y0, int* g)
{
    *g = gcd(fabs(a), fabs(b), x0, y0);
 
    // Condition if solution
    // does not exists
    if (c % *g) {
        return 0;
    }
 
    *x0 *= c / *g;
    *y0 *= c / *g;
 
    // Adjusting the sign
    // of x0 and y0
    if (a < 0)
        *x0 *= -1;
    if (b < 0)
        *y0 *= -1;
 
    return 1;
}
 
// Function to shift solution
void shift_solution(int* x, int* y,
                    int a, int b,
                    int shift_var)
{
 
    // Shifting to obtain
    // another solution
    *x += shift_var * b;
    *y -= shift_var * a;
}
 
// Function to find minimum
// value of x and y
int find_min_sum(int a, int b, int c)
{
    int x, y, g;
 
    // g is the gcd of a and b
    if (!possible_solution(a, b, c,
                        &x, &y, &g))
        return -1;
 
    a /= g;
    b /= g;
 
    // Store sign of a and b
    int sign_a = a > 0 ? +1 : -1;
    int sign_b = b > 0 ? +1 : -1;
 
    shift_solution(&x, &y, a, b, -x / b);
 
    // If x is less than 0, then
    // shift solution
    if (x < 0)
        shift_solution(&x, &y, a, b, sign_b);
 
    int minx1 = x;
 
    shift_solution(&x, &y, a, b, y / a);
 
    // If y is less than 0, then
    // shift solution
    if (y < 0)
        shift_solution(&x, &y, a, b, -sign_a);
 
    int minx2 = x;
 
    if (minx2 > x)
        swap(minx2, x);
    int minx = max(minx1, minx2);
 
    // Find intersection such
    // that both x and y are positive
 
    if (minx > x)
        return -1;
 
    // miny is value of y
    // corresponding to minx
    int miny = (c - a * x) / b;
 
    // Returns minimum value of x+y
    return (miny + minx);
}
 
// Driver Code
int main()
{
    // Given a, b, and c
    int a = 2, b = 2, c = 0;
 
    // Function Call
    cout << find_min_sum(a, b, c)
        << "\n";
 
    return 0;
}

Java




// Java program for the above approach
import java.lang.*;
class GFG{
     
public static int x = 0, y = 0,
                 x1 = 0, y1 = 0;
public static int x0 = 0, y0 = 0,
                   g = 0;
 
// Function to find the gcd of a & b
// by Euclid's method
 
// x and y store solution of
// equation ax + by = g                 
public static int gcd(int a, int b)
{
    if (b == 0)
    {
        x = 1;
        y = 0;
        return a;
    }
 
    int store_gcd = gcd(b, a % b);
 
    // Euclidean Algorithm
    x = y1;
    y = x1 - y1 * (a / b);
 
    // store_gcd returns
    // the gcd of a and b
    return store_gcd;
}
 
// Function to find any
// possible solution
public static int possible_solution(int a, int b,
                                    int c)
{
    g = gcd(Math.abs(a), Math.abs(b));
 
    // Condition if solution
    // does not exists
    if (c % g != 0)
    {
        return 0;
    }
 
    x0 *= c / g;
    y0 *= c / g;
 
    // Adjusting the sign
    // of x0 and y0
    if (a < 0)
        x0 *= -1;
    if (b < 0)
        y0 *= -1;
 
    return 1;
}
 
// Function to shift solution
public static void shift_solution(int a, int b,
                                  int shift_var)
{
     
    // Shifting to obtain
    // another solution
    x += shift_var * b;
    y -= shift_var * a;
}
 
// Function to find minimum
// value of x and y
public static int find_min_sum(int a, int b,
                               int c)
{
    int x = 0, y = 0, g = 0;
 
    // g is the gcd of a and b
    if (possible_solution(a, b, c) == 0)
        return -1;
 
    if (g != 0)
    {
        a /= g;
        b /= g;
    }
 
    // Store sign of a and b
    int sign_a = a > 0 ? + 1 : -1;
    int sign_b = b > 0 ? + 1 : -1;
 
    shift_solution(a, b, -x / b);
 
    // If x is less than 0, then
    // shift solution
    if (x < 0)
        shift_solution(a, b, sign_b);
 
    int minx1 = x;
 
    shift_solution(a, b, y / a);
 
    // If y is less than 0, then
    // shift solution
    if (y < 0)
        shift_solution(a, b, -sign_a);
 
    int minx2 = x;
 
    if (minx2 > x)
    {
        int temp = minx2;
        minx2 = x;
        x = temp;
    }
    int minx = Math.max(minx1, minx2);
 
    // Find intersection such
    // that both x and y are positive
 
    if (minx > x)
        return -1;
 
    // miny is value of y
    // corresponding to minx
    int miny = (c - a * x) / b;
 
    // Returns minimum value of x+y
    return (miny + minx);
}
 
// Driver Code
public static void main(String[] args)
{
     
    // Given a, b, and c
    int a = 2, b = 2, c = 0;
 
    // Function call
    System.out.println(find_min_sum(a, b, c));
}
}
 
// This code is contributed by grand_master

Python3




# Python3 program for the
# above approach
 
x, y, x1, y1 = 0, 0, 0, 0
x0, y0, g = 0, 0, 0
 
# Function to find the gcd
# of a & b by Euclid's method
 
# x and y store solution of
# equation ax + by = g
def gcd(a, b) :
     
    global x, y, x1, y1
    if (b == 0) :
     
        x = 1
        y = 0
        return a
     
    store_gcd = gcd(b, a % b)
     
    # Euclidean Algorithm
    x = y1
    y = x1 - y1 * (a // b)
     
    # store_gcd returns
    # the gcd of a and b
    return store_gcd
 
# Function to find any
# possible solution
def possible_solution(a, b, c) :
     
    global x0, y0, g
    g = gcd(abs(a), abs(b))
     
    # Condition if solution
    # does not exists
    if (c % g != 0) :
     
        return 0
     
    x0 *= c // g
    y0 *= c // g
     
    # Adjusting the sign
    # of x0 and y0
    if (a < 0) :
        x0 *= -1
    if (b < 0) :
        y0 *= -1
    return 1
 
# Function to shift solution
def shift_solution(a, b, shift_var) :
     
    global x, y
    # Shifting to obtain
    # another solution
    x += shift_var * b
    y -= shift_var * a
 
# Function to find minimum
# value of x and y
def find_min_sum(a, b, c) :
    global x, y, g
    x, y, g = 0, 0, 0
     
    # g is the gcd of a and b
    if (possible_solution(a, b, c) == 0) :
        return -1
     
    if (g != 0) :
     
        a //= g
        b //= g
     
    # Store sign of a and b
    if a > 0 :
        sign_a = 1
    else :
        sign_a = -1
     
    if b > 0 :
        sign_b = 1
    else :
        sign_b = -1
     
    shift_solution(a, b, -x // b)
     
    # If x is less than 0,
    # then shift solution
    if (x < 0) :
        shift_solution(a, b, sign_b)
     
    minx1 = x
     
    shift_solution(a, b, y // a)
     
    # If y is less than 0,
    # then shift solution
    if (y < 0) :
        shift_solution(a, b, -sign_a)
     
    minx2 = x
     
    if (minx2 > x) :
     
        temp = minx2
        minx2 = x
        x = temp
     
    minx = max(minx1, minx2)
     
    # Find intersection such
    # that both x and y are positive
    if (minx > x) :
        return -1
     
    # miny is value of y
    # corresponding to minx
    miny = (c - a * x) // b
     
    # Returns minimum value
    # of x + y
    return (miny + minx)
 
 
# Given a, b, and c
a, b, c = 2, 2, 0
 
# Function call
print(find_min_sum(a, b, c))
 
# This code is contributed by divyesh072019

C#




// C# program for the
// above approach
using System;
class GFG{
 
public static int x = 0, y = 0,
                  x1 = 0, y1 = 0;
public static int x0 = 0, y0 = 0,
                  g = 0;
 
// Function to find the gcd
// of a & b by Euclid's method
 
// x and y store solution of
// equation ax + by = g
public static int gcd(int a, int b)
{
  if (b == 0)
  {
    x = 1;
    y = 0;
    return a;
  }
 
  int store_gcd = gcd(b,
                      a % b);
 
  // Euclidean Algorithm
  x = y1;
  y = x1 - y1 *
      (a / b);
 
  // store_gcd returns
  // the gcd of a and b
  return store_gcd;
}
 
// Function to find any
// possible solution
public static int possible_solution(int a,
                                    int b,
                                    int c)
{
  g = gcd(Math.Abs(a),
          Math.Abs(b));
 
  // Condition if solution
  // does not exists
  if (c % g != 0)
  {
    return 0;
  }
 
  x0 *= c / g;
  y0 *= c / g;
 
  // Adjusting the sign
  // of x0 and y0
  if (a < 0)
    x0 *= -1;
  if (b < 0)
    y0 *= -1;
  return 1;
}
 
// Function to shift solution
public static void shift_solution(int a, int b,
                                  int shift_var)
{
  // Shifting to obtain
  // another solution
  x += shift_var * b;
  y -= shift_var * a;
}
 
// Function to find minimum
// value of x and y
public static int find_min_sum(int a,
                               int b,
                               int c)
{
  int x = 0, y = 0, g = 0;
 
  // g is the gcd of a and b
  if (possible_solution(a, b,
                        c) == 0)
    return -1;
 
  if (g != 0)
  {
    a /= g;
    b /= g;
  }
 
  // Store sign of a and b
  int sign_a = a > 0 ?
               +1 : -1;
  int sign_b = b > 0 ?
               +1 : -1;
 
  shift_solution(a, b,
                 -x / b);
 
  // If x is less than 0,
  // then shift solution
  if (x < 0)
    shift_solution(a, b,
                   sign_b);
 
  int minx1 = x;
 
  shift_solution(a, b,
                 y / a);
 
  // If y is less than 0,
  // then shift solution
  if (y < 0)
    shift_solution(a, b,
                   -sign_a);
 
  int minx2 = x;
 
  if (minx2 > x)
  {
    int temp = minx2;
    minx2 = x;
    x = temp;
  }
   
  int minx = Math.Max(minx1,
                      minx2);
 
  // Find intersection such
  // that both x and y are positive
  if (minx > x)
    return -1;
 
  // miny is value of y
  // corresponding to minx
  int miny = (c - a *
              x) / b;
 
  // Returns minimum value
  // of x + y
  return (miny + minx);
}
 
// Driver Code
public static void Main(String[] args)
{
  // Given a, b, and c
  int a = 2, b = 2, c = 0;
 
  // Function call
  Console.Write(find_min_sum(a, b, c));
}
}
 
// This code is contributed by Chitranayal

Javascript




<script>
 
    // Javascript program for the above approach
     
    let x = 0, y = 0, x1 = 0, y1 = 0;
    let x0 = 0, y0 = 0, g = 0;
 
    // Function to find the gcd
    // of a & b by Euclid's method
 
    // x and y store solution of
    // equation ax + by = g
    function gcd(a, b)
    {
      if (b == 0)
      {
        x = 1;
        y = 0;
        return a;
      }
 
      let store_gcd = gcd(b, a % b);
 
      // Euclidean Algorithm
      x = y1;
      y = x1 - y1 * parseInt(a / b, 10);
 
      // store_gcd returns
      // the gcd of a and b
      return store_gcd;
    }
 
    // Function to find any
    // possible solution
    function possible_solution(a, b, c)
    {
      g = gcd(Math.abs(a), Math.abs(b));
 
      // Condition if solution
      // does not exists
      if (c % g != 0)
      {
        return 0;
      }
 
      x0 *= parseInt(c / g, 10);
      y0 *= parseInt(c / g, 10);
 
      // Adjusting the sign
      // of x0 and y0
      if (a < 0)
        x0 *= -1;
      if (b < 0)
        y0 *= -1;
      return 1;
    }
 
    // Function to shift solution
    function shift_solution(a, b, shift_var)
    {
      // Shifting to obtain
      // another solution
      x += shift_var * b;
      y -= shift_var * a;
    }
 
    // Function to find minimum
    // value of x and y
    function find_min_sum(a, b, c)
    {
      let x = 0, y = 0, g = 0;
 
      // g is the gcd of a and b
      if (possible_solution(a, b, c) == 0)
        return -1;
 
      if (g != 0)
      {
        a = parseInt(a / g, 10);
        b = parseInt(b / g, 10);
      }
 
      // Store sign of a and b
      let sign_a = a > 0 ? +1 : -1;
      let sign_b = b > 0 ? +1 : -1;
 
      shift_solution(a, b, parseInt(-x / b, 10));
 
      // If x is less than 0,
      // then shift solution
      if (x < 0)
        shift_solution(a, b, sign_b);
 
      let minx1 = x;
 
      shift_solution(a, b, parseInt(y / a, 10));
 
      // If y is less than 0,
      // then shift solution
      if (y < 0)
        shift_solution(a, b, -sign_a);
 
      let minx2 = x;
 
      if (minx2 > x)
      {
        let temp = minx2;
        minx2 = x;
        x = temp;
      }
 
      let minx = Math.max(minx1, minx2);
 
      // Find intersection such
      // that both x and y are positive
      if (minx > x)
        return -1;
 
      // miny is value of y
      // corresponding to minx
      let miny = parseInt((c - a * x) / b, 10);
 
      // Returns minimum value
      // of x + y
      return (miny + minx);
    }
     
    // Given a, b, and c
    let a = 2, b = 2, c = 0;
 
    // Function call
    document.write(find_min_sum(a, b, c));
     
</script>

 
 

Output: 
0

 

 

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