Linear Diophantine equation which doesn't divide

Question

Im trying to find the solution for the linear Diophantine equation $55x + 22y = 400. $

I found $gcd(55,22) = 11$

therefore $11 = 55-22.2$ but 400 isnt a multiple of 11. is there any other way which i can find x and y or is it a dead end? Please help.
Thanks in advance.

Answer 1

It means that there is no solution.

Suppose on the contrary that there is a solution.

$$55x+22y = 400$$

then we have

$$11(5x+2y)=400$$ which means $11$ divides $400$, this is a contradiction.

Answer 2

This is based on a generalization theorem and you just have your way to prove it.

The theorem says:

For any Diophantine equation of the form ax+by = c, it is solvable if and only if gcd(a,b) divides c.

The proof will follow from your observation.