[Solution] The given equation is $[-77.66 x]=-78 x+1$, or $[(-78+0.34) x]=-78 x+1$.
Since $x$ is an integer, then $-78 x+[0.34 x]=-78 x+1$, which means $[0.34 x]=1$. Therefore, $1 \leqslant 0.34 x \leqslant 2$, which implies $3 \leqslant x \leqslant 5$.
$$
\therefore \quad x=3,4,5 \text {. }
$$

Therefore, the answer is $(C)$.
$33 \cdot 33[x]$ represents the greatest integer not exceeding $x$, then the number of integers $x$ that satisfy
$$
[-77.66 x]=[-77.66] x+1
$$

is
(A) 1 .
(B) 2 .
(C) 3 .
(D) 4 .
(Chinese Jiangsu Province Junior High School Mathematics Competition, 1987)