Given three integers a, b, and c representing a linear equation of the form: ax + by = c. The task is to find the initial integral solution of the given equation if a finite solution exists.
A Linear Diophantine equation (LDE) is an equation with 2 or more integer unknowns and the integer unknowns are each to at most degree of 1. Linear Diophantine equation in two variables takes the form of ax+by=c, where x,y are integer variables and a, b, c are integer constants. x and y are unknown variables.
Examples:
Input: a = 4, b = 18, c = 10
Output: x = -20, y = 5
Explanation: (-20)*4 + (5)*18 = 10
Input: a = 9, b = 12, c = 5
Output: No Solutions exists
Approach:
- First, check if a and b are non-zero.
- If both of them are zero and c is non-zero then, no solution exists. If c is also zero then infinite solution exits.
- For given a and b, calculate the value of x1, y1, and gcd using Extended Euclidean Algorithm.
- Now, for a solution to existing gcd(a, b) should be multiple of c.
- Calculate the solution of the equation as follows:
x = x1 * ( c / gcd ) y = y1 * ( c / gcd )
Below is the implementation of the above approach:
// C++ program for the above approach #include <bits/stdc++.h> using namespace std;
// Function to implement the extended // euclid algorithm int gcd_extend( int a, int b,
int & x, int & y)
{ // Base Case
if (b == 0) {
x = 1;
y = 0;
return a;
}
// Recursively find the gcd
else {
int g = gcd_extend(b,
a % b, x, y);
int x1 = x, y1 = y;
x = y1;
y = x1 - (a / b) * y1;
return g;
}
} // Function to print the solutions of // the given equations ax + by = c void print_solution( int a, int b, int c)
{ int x, y;
if (a == 0 && b == 0) {
// Condition for infinite solutions
if (c == 0) {
cout
<< "Infinite Solutions Exist"
<< endl;
}
// Condition for no solutions exist
else {
cout
<< "No Solution exists"
<< endl;
}
}
int gcd = gcd_extend(a, b, x, y);
// Condition for no solutions exist
if (c % gcd != 0) {
cout
<< "No Solution exists"
<< endl;
}
else {
// Print the solution
cout << "x = " << x * (c / gcd)
<< ", y = " << y * (c / gcd)
<< endl;
}
} // Driver Code int main( void )
{ int a, b, c;
// Given coefficients
a = 4;
b = 18;
c = 10;
// Function Call
print_solution(a, b, c);
return 0;
} |
// Java program for the above approach class GFG{
static int x, y;
// Function to implement the extended // euclid algorithm static int gcd_extend( int a, int b)
{ // Base Case
if (b == 0 )
{
x = 1 ;
y = 0 ;
return a;
}
// Recursively find the gcd
else
{
int g = gcd_extend(b, a % b);
int x1 = x, y1 = y;
x = y1;
y = x1 - (a / b) * y1;
return g;
}
} // Function to print the solutions of // the given equations ax + by = c static void print_solution( int a, int b, int c)
{ if (a == 0 && b == 0 )
{
// Condition for infinite solutions
if (c == 0 )
{
System.out.print( "Infinite Solutions " +
"Exist" + "\n" );
}
// Condition for no solutions exist
else
{
System.out.print( "No Solution exists" +
"\n" );
}
}
int gcd = gcd_extend(a, b);
// Condition for no solutions exist
if (c % gcd != 0 )
{
System.out.print( "No Solution exists" + "\n" );
}
else
{
// Print the solution
System.out.print( "x = " + x * (c / gcd) +
", y = " + y * (c / gcd) + "\n" );
}
} // Driver Code public static void main(String[] args)
{ int a, b, c;
// Given coefficients
a = 4 ;
b = 18 ;
c = 10 ;
// Function Call
print_solution(a, b, c);
} } // This code is contributed by Rajput-Ji |
# Python3 program for the above approach import math
x, y = 0 , 0
# Function to implement the extended # euclid algorithm def gcd_extend(a, b):
global x, y
# Base Case
if (b = = 0 ):
x = 1
y = 0
return a
# Recursively find the gcd
else :
g = gcd_extend(b, a % b)
x1, y1 = x, y
x = y1
y = x1 - math.floor(a / b) * y1
return g
# Function to print the solutions of # the given equations ax + by = c def print_solution(a, b, c):
if (a = = 0 and b = = 0 ):
# Condition for infinite solutions
if (c = = 0 ):
print ( "Infinite Solutions Exist" )
# Condition for no solutions exist
else :
print ( "No Solution exists" )
gcd = gcd_extend(a, b)
# Condition for no solutions exist
if (c % gcd ! = 0 ):
print ( "No Solution exists" )
else :
# Print the solution
print ( "x = " , int (x * (c / gcd)), ", y = " , int (y * (c / gcd)), sep = "")
# Given coefficients a = 4
b = 18
c = 10
# Function Call print_solution(a, b, c) # This code is contributed by decode2207. |
// C# program for the above approach using System;
class GFG{
static int x, y;
// Function to implement the extended // euclid algorithm static int gcd_extend( int a, int b)
{ // Base Case
if (b == 0)
{
x = 1;
y = 0;
return a;
}
// Recursively find the gcd
else
{
int g = gcd_extend(b, a % b);
int x1 = x, y1 = y;
x = y1;
y = x1 - (a / b) * y1;
return g;
}
} // Function to print the solutions of // the given equations ax + by = c static void print_solution( int a, int b, int c)
{ if (a == 0 && b == 0)
{
// Condition for infinite solutions
if (c == 0)
{
Console.Write( "Infinite Solutions " +
"Exist" + "\n" );
}
// Condition for no solutions exist
else
{
Console.Write( "No Solution exists" +
"\n" );
}
}
int gcd = gcd_extend(a, b);
// Condition for no solutions exist
if (c % gcd != 0)
{
Console.Write( "No Solution exists" + "\n" );
}
else
{
// Print the solution
Console.Write( "x = " + x * (c / gcd) +
", y = " + y * (c / gcd) + "\n" );
}
} // Driver Code public static void Main(String[] args)
{ int a, b, c;
// Given coefficients
a = 4;
b = 18;
c = 10;
// Function call
print_solution(a, b, c);
} } // This code contributed by amal kumar choubey |
<script> // Javascript program for the above approach let x, y; // Function to implement the extended // euclid algorithm function gcd_extend(a, b)
{ // Base Case
if (b == 0)
{
x = 1;
y = 0;
return a;
}
// Recursively find the gcd
else
{
let g = gcd_extend(b, a % b);
let x1 = x, y1 = y;
x = y1;
y = x1 - Math.floor(a / b) * y1;
return g;
}
} // Function to print the solutions of // the given equations ax + by = c function print_solution(a, b, c)
{ if (a == 0 && b == 0)
{
// Condition for infinite solutions
if (c == 0)
{
document.write( "Infinite Solutions " +
"Exist" + "\n" );
}
// Condition for no solutions exist
else
{
document.write( "No Solution exists" +
"\n" );
}
}
let gcd = gcd_extend(a, b);
// Condition for no solutions exist
if (c % gcd != 0)
{
document.write( "No Solution exists" + "\n" );
}
else
{
// Print the solution
document.write( "x = " + x * (c / gcd) +
", y = " + y * (c / gcd) + "<br/>" );
}
} // Driver Code let a, b, c;
// Given coefficients
a = 4;
b = 18;
c = 10;
// Function Call
print_solution(a, b, c);
</script> |
x = -20, y = 5
Time Complexity: O(log(max(A, B))), where A and B are the coefficient of x and y in the given linear equation.
Auxiliary Space: O(1)