How can I find the integer solutions of $17x+13y=282$?

Question

I am trying to find a reliable fast way to find the integer solutions for equations such as $17x+13y=282$. I need to know as fast as possible if such equation has only 1 pair of integers $x,y$ that meet the equation, or more than one.

What would be the best method to approach such a problem? Thank you.

Ps. Assume I have no calculator. Just pen and paper.

Answer 1

This is a linear Diophantine equation. The techniques for solving are well-known. If there is one solution, then there are infinitely many, so there is no case in which there is just one solution.

Since the $\gcd(17,13) =1,$ first find a solution to $17x+13y = 1$. Then multiply the solution by $282$. Then whatever the solution is, you can add any multiple of $13$ to $x$ and subtract the same multiple of $17$ from $y$ and get another solution.

Answer 2

$4\cdot 13 = 52\\ 3\cdot 17 = 51\\ 4\cdot 13 - 3\cdot 17 = 1$

$x = -282(3), y = 282(4)$ is a solution.

$x = -846 + 13k, y = 1128 - 17k$

If you want to add the additional criteria that $x,y \in \mathbb N$

then $\frac {846}{13} = 65 \frac 1{13}$ and $\frac {1128}{17} = 66 \frac {6}{17}$

Then there is only one $k = 66$ that satisfies the equation.

Answer 3

Note that $$ 4(13) -3(17)=1$$

Therefore $$ 4(282)(13)-3(282)(17)=282 $$

Thus $x= -3(282)=-846 $ and $y= 4(282)=1128$