Example 1 Arrange the simplest fractions with denominators not exceeding 99 between $[0,1]$ in ascending order, find the two numbers adjacent to $\frac{17}{76}$.
Let $x, y \in \mathbf{N}^{*}, (x, y)=1$, and $\frac{x}{y}$ is the number to the left of $\frac{17}{76}$ in the above sequence, then
$$\frac{17}{76}-\frac{x}{y}=\frac{17 y-76 x}{76 y}>0$$

Notice that $17 y-16 x$ is an integer, so $17 y-76 x \geqslant 1$.
We first solve the indeterminate equation
$$17 y-76 x=1$$

for positive integer solutions $(x, y)$ satisfying $1 \leqslant y \leqslant 99$.
We can use the following method to find a particular solution to (3):
$$y=4 x+\frac{8 x+1}{17} \in \mathbf{Z}$$

By trial, we find that $(x, y)=(2,9)$ is a particular solution. Therefore, all integer solutions to (3) are
$$\left\{\begin{array}{l}
x=2+17 t \\
y=9+76 t
\end{array} t \in \mathbf{Z}\right.$$

Among the positive integer solutions to (3), $(x, y)=(19,85)$ is the solution that satisfies $1 \leqslant y \leqslant 99$ and has the largest $y$, and at this point $y=85>\frac{99}{2}$, so the number to the left of $\frac{17}{76}$ is $\frac{19}{85}$.

Similarly, the number to the right of $\frac{17}{76}$ is $\frac{15}{67}$.