Find all solutions to the equation $101x + 97y = 2.$

Find all solutions to the equation $$101x + 97y = 2.$$ Supposed answer is $$x = -24 +97k$$, $$y = 25-101k$$. I am unsure what steps to take to find these solutions.

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The correct form of your equation is $$101x +97y=1$$ if the solutions are correct. You must use homogeneous equation for general solutions; if equation is $$ax+by=c$$ then it's homogeneous equation is $$ax+by=0$$. the solutions of this equation are: $$x=b$$ and $$y=-a$$, so the general form of solutions is:

$$x=bk+x_1$$

$$y=-ak+y_1$$

Where $$x_1$$ and $$y_1$$ are one solution of initial equation $$101 x +97y=1$$. you can see that $$x_1=-24$$ and $$y_1=25$$ are one solution, so general solutions are:

$$x=97 k -24$$

$$y=-101 k +25$$

Finding primitive solutions: $$101x +97y=1$$ indicates that $$101 x$$ and $$97 y$$ are two consecutive numbers with opposite signs; One odd and it's absolute value greater than the other which is even. Suppose:

$$101 x=2k$$

$$97 y=2k+1$$

$$97 y-101 x=1$$

$$97 y +101 x= 4k+1$$

We rewrite this relation as:

$$(4\times 24+1) y +(4\times 25+1)x=4(24 y+ 25 x) +x+y=4k+1$$

We may assume:

$$24y+25x=k$$; which homogeneous equation is $$24y+25x=0$$ and gives $$y=25, x=-24$$ and $$x+y=1$$ which is also satisfied with these values of x and y. So $$x=-24, y=25$$ can be accepted as primitive solutions.

• The answer is more or less trivial without stating how to find solutions to the initial condition. – Szeto Sep 27 '18 at 9:01