Example 1 Solve the congruence equation $589 x \equiv 1026(\bmod 817)$.
Solve $589 x \equiv 1026(\bmod 817) \stackrel{(\mathrm{i})}{\Longleftrightarrow}-228 x \equiv 209(\bmod 81)$

This indicates that the last congruence equation about $u$ has 19 solutions modulo 19:
$$u \equiv 0,1,2, \cdots, 18(\bmod 19)$$

According to (iii), i.e., equation (13), tracing back step by step, we get: the congruence equation about $w$ modulo 38 has 19 solutions
$$w \equiv(38 u+19) / 19 \equiv 2 u+1(\bmod 38), \quad u=0,1, \cdots, 18$$

The congruence equation about $z$ modulo 95 has 19 solutions:
$$z \equiv(95 w-19) / 38 \equiv 5 u+2(\bmod 95), \quad u=0,1, \cdots, 18$$

The congruence equation about $y$ modulo 228 has 19 solutions:
$$y \equiv(228 z+19) /(-95) \equiv-12 u-5(\bmod 228), \quad u=0,1, \cdots, 18$$

Finally, we get the congruence equation about $x$ modulo 817 has 19 solutions:
$$\begin{array}{c}
x \equiv(817 y+209) /(-228) \equiv 43 u+17(\bmod 817) \\
u=0,1, \cdots, 18
\end{array}$$
$$\begin{array}{l}
\stackrel{\text { (ii) }}{\Longleftrightarrow} 817 y \equiv-209(\bmod 228) \stackrel{(\mathrm{i})}{\Longleftrightarrow}-95 y \equiv 19(\bmod 228) \\
\stackrel{\text { (ii) }}{\Longleftrightarrow} 228 z \equiv-19(\bmod 95) \stackrel{(\mathrm{i})}{\Longleftrightarrow} 38 z \equiv-19(\bmod 95) \\
\stackrel{\text { (ii) }}{\Longleftrightarrow} 95 w \equiv 19(\bmod 38) \stackrel{(\text { i) }}{\Longleftrightarrow} 19 w \equiv 19(\bmod 38) \\
\stackrel{\text { (ii) }}{\Longleftrightarrow} 38 u \equiv-19(\bmod 19) \stackrel{(\mathrm{i})}{\Longleftrightarrow} \cdot u \equiv 0(\bmod 19) \text {. }
\end{array}$$