4. (Kyiv City Math Competition Question) (1) In which base is 16324 the square of 125?
(2) In which base is $4 \cdot 13=100$?
4. (1) Let the base of the numeral system be $b$. From the problem, we know that $b>6$. At this point, we can obtain the equation $b^{4}+6 b^{3}+3 b^{2}+2 b+4=\left(b^{2}+2 b+5\right)^{2}$. Simplifying, we get $2 b^{3}-11 b^{2}-18 b-21=0$, which can be factored as $(b-7)\left(2 b^{2}+3 b+3\right)=0$. This equation has a unique real root $b=7$, so in the base-7 numeral system, 16324 is the square of 125.
(2) Let the base of the numeral system be $b$. From the problem, we know that $b>4$. At this point, we can obtain the equation $4(1 \cdot b+3)=1 \cdot b^{2}+0 \cdot b+0$, and $b>4$, thus $b^{2}-4 b-12=0$. Solving for the positive value of $b$, we get $b=6$, so the given equation is in the base-6 numeral system.