Side 3. Find the general integer solution of the equation $41 x+177 y=4$.

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Solution:
$3\left|\begin{array}{rr}41 & 177 \\ 39 & 164 \\ \hline 2 & 13 \\ 2 & 12 \\ \hline 0 & 1\end{array}\right| 6$

It is clear that $177=4 \times 41+13,13=177-4 \times 41$,
$$
\begin{array}{ll}
41=3 \times 13+2, & 2=41-3 \times 13, \\
13=6 \times 2+1, & 1=13-6 \times 2 .
\end{array}
$$

To express 1 using 41 and 177, i.e., in the form $1=41 \cdot(\quad)+177 \cdot(\quad)$, substituting (3) we get
$$
\begin{aligned}
1 & =13-6 \times(41-3 \times 13) \\
& =19 \times 13-6 \times 41 .
\end{aligned}
$$

Substituting (i) into (4) we get
$$
\begin{aligned}
1 & =19 \times(177-4 \times 41)-6 \times 41 \\
& =41 \times(-82)+177 \times 19,
\end{aligned}
$$

i.e., $41 \times(-82)+177 \times 19=1$.
Thus, $x_{0}=-82, y_{0}=19$ is a solution to the equation $41 x+177 y=1$.
Therefore,
$$
\begin{array}{l}
x_{1}=4 \times(-82)=-328, \\
y_{1}=4 \times 19=76
\end{array}
$$

is a particular solution to the original equation, and the general solution to the original equation is
$$
\begin{array}{l}
x=-328+177 t, \\
y=76-41 t .(t \text { is an integer) }
\end{array}
$$