Given the following diophantine equation: $53x + 12y = 2$ determine the integer solutions (if any).

Question

The question that I am working is:

Given the following diophantine equation: $53x + 12y = 2$ determine the integer solutions (if any).

The problem that I am facing is that I tried to find two solutions but keep getting an incorrect $x_{0}$ and $y_{0}$ values.

Here is my work: Claim - "Yes, integer solutions do exists."

Using Euclid's Algorithm:

$53 = 12(4) + 5$
$12 = 5(2) + 2$
$5 = 2(2) + 1$
$1 = 1(1) + 0$

Hence, $\text{gcd}(53,12) = 1$

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System of equations:

$1 = 2 - 1(1)$
$1 = 5 - 2(2)$
$2 = 12 - 5(2)$
$5 = 53 - 12(4)$

Answer 1

Solution of Linear Congruence says solution always exists as $2$ is divisible by $(12,53)=1$

$$53x+12y=2=12\cdot9-2\cdot53$$

$$\iff53(x+2)=12(9-y)\iff\dfrac{53(x+2)}{12}=9-y$$ which is an integer

$\implies12|53(x+2)\iff12|(x+2)$ as $(12,53)=1$

$\implies x=12m-2$ where $m$ is any integer

Put this in $53x+12y=2$

Answer 2

If you just want to use basic algebra and common sense, take a look at the equation. Clearly $12y$ and $2$ are even so we have to have $53x$ even, so $x$ is even. A quick look and $x = -2$ will give $y = 9$, and $x = 10$ will give $y = -44$, now if you get the hang of it, keep on adding or subtracting $12$ to/from $x$ and other solutions will follow. Of course lab bhattacharjee's solution is more complete and rigorous, but this is one way such things can be dealt with intuitively.