A quick method to solve $89y-273x=40$

Question

how to solve this equation $$89y-273x=40$$

I saw this question somewhere and this obviously can be solved by hit and trial but is there an easier method to solve it, something more definite? I need an integral solution.

Answer 1

I think you mean integer solutions? So $x,y \in \mathbb{N}$? If so, you will find all the solutions by applying the euclidean algorithm. First you are calculating the greates common divisor with it, and then, going back, you will find a solution for the equation $89y -273x = gcd(89,273)=1$. Now, by multpiplying 40 on both sites you get your equation with your solutions.

Answer 2

From the Euclidean Algorithm we have that $$\gcd(89,273)=1$$ Thus there are infinitely many pairs of $(x,y)$ integers (solutions) that satisfy it.

Indeed, since there are $a,b \in\Bbb{Z}$, $a,b$ not unique*, such that $$89a+273b=1\Rightarrow\\40(89a+273b)=40\Rightarrow\\89(40a)+273(40b)=40$$

*To see that $a,b$ are not unique, simply consider $a'=a-273b, b'=b+89b\Rightarrow$ $$89a'+273b'=89a-89*273b+273b+89*273b=89a+273b=1$$