$1 \cdot 167$ Find all such three-digit numbers $\overline{a b c}$, the last three digits of whose square are also $a b c$.

Find all such three-digit numbers $\overline{a b c}$, the last three digits of whose square are also $a b c$.
[Solution] Let the three-digit number that satisfies the requirement be $A$, then $A^{2}-A=A(A-1)$ is divisible by 1000 $=8 \cdot 125$. Since $A$ and $A-1$ are coprime, it can only be that 125 divides $A$ and 8 divides $A-1$, or 125 divides $A-1$ and 8 divides $A$. It is easy to verify that the only three-digit numbers that satisfy the requirement are 376 or 625.