Solve $56x+63y=1$

Question

Solve $56x+63y=1$

What I did:

$$\gcd(56,63):$$

$$\qquad63=56\cdot 1+7\\56=7\cdot 8$$

$$\Longrightarrow \gcd=7$$

Since $7\nmid 1$ there is no solution, but I think that Wolfram says something else

Answer 1

There aren't integer solutions, in other words you cannot find a couple of integer satisfying the equation. However that equation describe a line on the plane, so if you are looking for rational or real solution, these are infinite.

Answer 2

"In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied"

See Linear Congruence Theroem

Answer 3

The equation can be rewritten as $7(8x+9y)=1$ so $$8x+9y = \frac{1}{7}.$$ Whether this equation has any solutions depends on from which domain you take $x$ and $y$ to be. The way your question is written sort of implies that you want $x,y$ to be whole numbers. Then the above equation immediately implies that there won't be any solutions since $8x+9y$ will always be a whole number and $\frac{1}{7}$ isn't. This same argument can be extended to show that equations of the form $$ a x + b y = 1 $$ have no integer solutions for $x,y$ if $\gcd(a,b)>1$. On the other hand, the extended Euclidean algorithm shows that you can always find integers $x,y$ such that $$ ax+by = \gcd(a,b). $$ Wolfram interprets the domain of $x,y$ to be the real numbers. In this case there are an infinite amount of solutions to the equation and they represent a line in the plane.