I have the following equation to which I'm supposed to find all integer solutions: $$25x-14y=20$$
I'm ignoring the negative sign for now, so:
$a=25,$ $b=14$
$25=14\times1+11$
$14=11\times1+3$
$11=3\times3+2$
$3=2\times1+1$
$d=\gcd(25,14)=1$
$1=3-2$
$1=4\times3-11$
$1=4\times14-5\times11$
$1=9\times14-5\times25$
$20\times1=20(9\times14-5\times25)$
$20=180\times14-100\times25$
$x0=-100$
$y0=-180$ Here I compensate for the negative sign I left out at the beginning, correct me if I'm wrong.
As far as I know the solutions should be found using the following formula:
$x=x0+\frac{b}{d}n$
$y=y0-\frac{a}{d}n$
$x=-100+14n$
$y=-180-25n$
But the answer says it should be $y=-180+25n$ so I guess there's something I either misunderstood or missed by mistake.
Thanks!
The sets $\{-180+25n\,|\,n\in\mathbb{Z}\}$ and $\{-180-25n\,|\,n\in\mathbb{Z}\}$ are the same.