Five. (25 points) Given three integers $a, b, c$ whose sum is 13, and $\frac{b}{a} = \frac{c}{b}$. Find the maximum and minimum values of $a$, and the corresponding values of $b$ and $c$ at these times.
Five, let $\frac{b}{a}=\frac{c}{b}=x$, then $b=a x, c=a x^{2}$.
Thus, $a+b+c=13$ becomes $a\left(x^{2}+x+1\right)=13$.
Since $a \neq 0$, we have
$$
x^{2}+x+1-\frac{13}{a}=0 \text {. }
$$

Also, since $a, b, c$ are integers, the solutions to equation (1) must be rational numbers, i.e., $\Delta=1-4\left(1-\frac{13}{a}\right)=\frac{52}{a}-3>0$.
Solving this, we get $1 \leqslant a<\frac{52}{3}$, and $\sqrt{\Delta}$ must be a rational number.
Thus, $1 \leqslant a \leqslant 16$.
When $a=1$, equation (1) becomes $x^{2}+x-12=0$.
Solving this, we get $x_{1}=-4, x_{2}=3$.
Thus, $a_{\min }=1, b=-4, c=16 ; a_{\min }=1, b=3, c=9$.
When $a=16$, equation (1) becomes $x^{2}+x+\frac{3}{16}=0$.
Solving this, we get $x_{1}=-\frac{3}{4}, x_{2}=-\frac{1}{4}$.
Thus, $a_{\max }=16, b=-12, c=9 ; a_{\max }=16, b=-4, c=1$.
(Tang Xianjiang provided)