How to find the first lattice point in the first quadrant on the line $21x-101y=1$?

Question

How to find the first lattice point in the first quadrant on the line $21x-101y=1$?

I can find the lattice point with the help of modular arithmetic. But is there any simple way to do that?

Answer 1

\begin{align*} &\text{$x$ is an integer such that $21x - 101y = 1$, for some integer $y$}\\[4pt] \iff\;&21x \equiv 1\;(\text{mod}\;101)\\[4pt] \iff\;&105x \equiv 5\;(\text{mod}\;101)\\[4pt] \iff\;&4x \equiv 5\;(\text{mod}\;101)\\[4pt] \iff\;&100x \equiv 125\;(\text{mod}\;101)\\[4pt] \iff\;&-x \equiv 24\;\;(\text{mod}\;101)\\[4pt] \iff\;&x \equiv -24\;\;(\text{mod}\;101)\\[4pt] \iff\;&x \equiv 77\;\;(\text{mod}\;101)\\[4pt] \iff\;&x = 77+101k,\;\text{for some integer $k$}\\[4pt] \end{align*} It follows that the smallest nonnegative qualifying value of $x$ is $77$.

Plugging $x=77$ into the equation $21x-101y=1$ yields $y=16$, so the required point is $(x,y) = (77,16)$.