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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.

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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.

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  • $\begingroup$ Thanks for the edit. $\endgroup$ – Siong Thye Goh Oct 16 '18 at 7:35
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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.

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