Example 12: When divided by 7, the remainder is 1; when divided by 8, the remainder is 1; when divided by 9, the remainder is 3. Find the number. (Translation: Find a positive integer $x$ such that when $x$ is divided by 7, the remainder is 1; when $x$ is divided by 8, the remainder is 1; and when $x$ is divided by 9, the remainder is 3.)
According to the problem, we have
\( x \equiv 1 \pmod{7}, x \equiv 1 \pmod{8}, x \equiv 3 \pmod{9} \).
In Theorem 1, let \( m_1 = 7, m_2 = 8, m_3 = 9, b_1 = b_2 = 1, b_3 = 3 \); at this point, we have
\[
\begin{array}{l}
M = 7 \times 8 \times 9 = 504 \\
M_1 = \frac{504}{7} = 72 \\
M_2 = \frac{504}{8} = 63 \\
M_3 = \frac{504}{9} = 56
\end{array}
\]

Let \( M_1' \) be a positive integer that satisfies \( M_1' M_1 \equiv 1 \pmod{7} \), then we have \( 1 \equiv M_1' M_1 \equiv 72 M_1' \equiv 2 M_1' \pmod{7} \), so we get \( M_1' = 4 \). Let \( M_2' \) be a positive integer that satisfies \( M_2' M_2 \equiv 1 \pmod{8} \), then we have \( 1 \equiv M_2' M_2 \equiv 63 M_2' \equiv 7 M_2' \pmod{8} \), so \( M_2' = 7 \). Let \( M_3' \) be a positive integer that satisfies \( M_3' M_3 \equiv 1 \pmod{9} \), then we have \( 1 \equiv M_3' M_3 \equiv 56 M_3' \equiv 2 M_3' \pmod{9} \), so \( M_3' = 5 \). Therefore, by (46), we get \( x \equiv 72 \times 4 + 7 \times 63 + 3 \times 5 \times 56 \equiv 57 \pmod{504} \), that is
\[
x = 57 + 504k, \quad k = 0, 1, 2, \cdots.
\]