Example 19 (2005 Croatian Mathematical Competition Problem) Find all positive integers that can be represented in decimal as $13 x y 45 z$, and are divisible by 792, where $x, y, z$ are unknowns.
Solution: Since $792=8 \times 9 \times 11$, the number $13 x y 45 z$ is divisible by $8,9,11$. From $8 \mid \overline{13 x y 45 z}$, we know $8 \mid \overline{45 z}$.
And $\overline{45 z}=450+z=448+(z+2)$, thus, $8 \mid(z+2)$.
Therefore, $z=6$.
From $9 \mid \overline{13 x y 456}$, we know
$9 \mid(1+3+x+y+4+5+6)$.
And $1+3+x+y+4+5+6=x+y+19=18+(x+y+1)$,
thus, $9 \mid(x+y+1)$.
Therefore, $x+y=8$ or $x+y=17$.
From $11 \mid \overline{13 x y 456}$, we know
$11 \mid(6-5+4-y+x-3+1)$.
And $6-5+4-y+x-3+1=x-y+3$, thus,
$x-y=-3$ or $x-y=8$.
At this point, there are two possibilities:
(1) If $x+y$ is even, then $x+y=8$ and $x-y=8$. Thus, $x=8, y=0$.
(2) If $x+y$ is odd, then $x+y=17$ and $x-y=-3$. Thus, $x=7, y=10$, which is impossible.
Therefore, the only positive integer is 1380456.
Example 20 (2003 Canadian Mathematical Olympiad) Find the last three digits of $2003^{2002^{2001}}$.
Solution $2003^{2002^{2001}} \equiv 3^{2002^{2001}}\left(\bmod 10^{3}\right)$
$$
\begin{array}{l}
=9^{2^{2000} \times 1001^{2001}} \\
=(10-1)^{k}\left(\text { let } k=2^{2000} \times 1001^{2001}\right) \\
=C_{k}^{2} \times 10^{2}-k \times 10+1\left(\bmod 10^{3}\right) .
\end{array}
$$

Where $k=2^{2000} \times 1001^{2001} \equiv 2^{2000}=1024^{200}$
$$
\begin{array}{l}
\equiv 24^{200}=3^{200} \times 2^{600} \equiv 3^{200} \times 24^{60}=3^{260} \times 2^{180} \\
\equiv 3^{260} \times 24^{18}=3^{278} \times 2^{54} \\
\equiv 3^{278} \times 24^{5} \times 2^{4}=3^{283} \times 2^{19} \\
\equiv 3^{283} \times 24 \times 2^{9}=3^{284} \times 2^{12} \\
\equiv 3^{284} \times 24 \times 2^{2}=(10-1)^{142} \times 96 \\
\equiv\left(C_{142}^{2} \times 10^{2}-142 \times 10+1\right) \times 96 \\
\equiv 681 \times 96 \equiv 376\left(\bmod 10^{3}\right) .
\end{array}
$$

Thus, $2003^{2002^{2001}} \equiv C_{376}^{2} \times 10^{2}-376 \times 10+1$
$$
\equiv 241\left(\bmod 10^{3}\right) \text {. }
$$