57. Given an integer $a$, representing a leg of a right triangle, all sides of which are measured in integers; determine the length of the other leg $b$ and the hypotenuse $c$. How many solutions are there? Solve the problem, for example, for $a=15$ and for $a=12$.
57. $c^{2}-b^{2}=a^{2}$ or $(c+b)(c-b)=a^{2}$.

$a=15 \quad(c+b)(c-b)=15^{2}=3 \cdot 3 \cdot 5 \cdot 5$.

1) $(c+b)(c-b)=225 \cdot 1$

$c+b=225$

$c-b=1$
$c=113, b=112$

3) $(c+b)(c-b)=45 \cdot 5$

$c+b=45$
$c-b=5$
$c=25, b=20$
2) $(c+b)(c-b)=75 \cdot 3$

$c+b=75$

$\frac{c-b=3}{c=39, b=36}$

4) $(c+b)(c-b)=25 \cdot 9$

$c+b=25$

$\frac{c-b=9}{c=17, b=8}$