11.27 In what base, 16324 is the square of 125.
(Kyiv Mathematical Olympiad, 1954)
［Solution］Let the base of the numeral system be $b$.
By the problem, it is clear that $\quad b>6$.
At this point, we can obtain the equation
$$
b^{4}+6 b^{3}+3 b^{2}+2 b+4=\left(b^{2}+2 b+5\right)^{2},
$$

After rearrangement, we get
$$
\begin{array}{c}
2 b^{3}-11 b^{2}-18 b-21=0, \\
(b-7)\left(2 b^{2}+3 b+3\right)=0 .
\end{array}
$$

This equation has only one real root $b=7$.
Thus, in base 7, 16324 is the square of 125.