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Solution of Linear Diophantine Equation

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How to find all solutions of Linear Diophantine Equation $a cdot x + b cdot y = c $ given $a,b,c$ where $c$ is divisible by $gcd(a,b)$ and constraints are $x_0 leq x leq x_1$ and $y_0 leq y leq y_1$ ?

discrete-mathematics constraints linear-diophantine-equations

share|cite|improve this question

edited Mar 30 at 13:45

Parth Patel

asked Mar 29 at 9:44

Parth PatelParth Patel

367

$endgroup$

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  Solution set of $(x,y,c)$ is every $x,y$ within the constraint, since every linear combination of $a,b$ is divisible by their GCD.
  $endgroup$
  – L KM
  Mar 29 at 10:03
  

  
  
  
  


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  $begingroup$
  @LKM. Given a = 1, b = c = 2, and 0 <= x,y <= 1000, how many integers x,y have x + 2y = 2???
  $endgroup$
  – William Elliot
  Mar 30 at 2:22
  

  
  
  
  


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  $begingroup$
  Then of course there is 2 solutions, namely, $(x,y)=(0,1), (2,0)$. So the question should be formulated as follows? \ How to find all solutions (x,y) of Linear Diophantine Equation $a cdot x + b cdot y = c$ with $a,b,c$ fixed and c divisible by $gcd(a,b)$ ?
  $endgroup$
  – L KM
  Mar 30 at 4:10
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Edited the question.
  $endgroup$
  – Parth Patel
  Mar 30 at 13:45
  

  
  
  
  

add a comment | 

0

$begingroup$

How to find all solutions of Linear Diophantine Equation $a cdot x + b cdot y = c $ given $a,b,c$ where $c$ is divisible by $gcd(a,b)$ and constraints are $x_0 leq x leq x_1$ and $y_0 leq y leq y_1$ ?

discrete-mathematics constraints linear-diophantine-equations

share|cite|improve this question

edited Mar 30 at 13:45

Parth Patel

asked Mar 29 at 9:44

Parth PatelParth Patel

367

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Solution set of $(x,y,c)$ is every $x,y$ within the constraint, since every linear combination of $a,b$ is divisible by their GCD.
  $endgroup$
  – L KM
  Mar 29 at 10:03
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @LKM. Given a = 1, b = c = 2, and 0 <= x,y <= 1000, how many integers x,y have x + 2y = 2???
  $endgroup$
  – William Elliot
  Mar 30 at 2:22
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Then of course there is 2 solutions, namely, $(x,y)=(0,1), (2,0)$. So the question should be formulated as follows? \ How to find all solutions (x,y) of Linear Diophantine Equation $a cdot x + b cdot y = c$ with $a,b,c$ fixed and c divisible by $gcd(a,b)$ ?
  $endgroup$
  – L KM
  Mar 30 at 4:10
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Edited the question.
  $endgroup$
  – Parth Patel
  Mar 30 at 13:45
  

  
  
  
  

add a comment | 

$begingroup$

How to find all solutions of Linear Diophantine Equation $a cdot x + b cdot y = c $ given $a,b,c$ where $c$ is divisible by $gcd(a,b)$ and constraints are $x_0 leq x leq x_1$ and $y_0 leq y leq y_1$ ?

discrete-mathematics constraints linear-diophantine-equations

share|cite|improve this question

edited Mar 30 at 13:45

Parth Patel

asked Mar 29 at 9:44

Parth PatelParth Patel

367

$endgroup$

share|cite|improve this question

edited Mar 30 at 13:45

Parth Patel

asked Mar 29 at 9:44

Parth PatelParth Patel

367

share|cite|improve this question

edited Mar 30 at 13:45

Parth Patel

asked Mar 29 at 9:44

Parth PatelParth Patel

367

asked Mar 29 at 9:44

Parth PatelParth Patel

367

asked Mar 29 at 9:44

Parth PatelParth Patel

367