123. How many right-angled triangles are there with sides of integer lengths, if one of the legs of these triangles is equal to 15?
Solution. If $x$ is the hypotenuse and $y$ is the unknown leg of a right triangle, then $x^{2}-y^{2}=15^{2} ;(x-y)(x+y)=$ $=3 \cdot 3 \cdot 5 \cdot 5$

Since $(x-y)$ and $(x+y)$ are natural numbers, and $x+y > x-y$, there are only 4 cases:

$$
\left\{\begin{array} { l } 
{ x - y = 1 } \\
{ x + y = 225 }
\end{array} \left\{\begin{array} { l } 
{ x - y = 3 } \\
{ x + y = 75 }
\end{array} \quad \left\{\begin{array}{l}
x-y=5 \\
x+y=45
\end{array} ;\left\{\begin{array}{l}
x-y=9 \\
x+y=25
\end{array}\right.\right.\right.\right.
$$

By solving these 4 systems of equations, we find that there are 4 right triangles that satisfy the given property. Their sides are: 1) $15 ; 112 ; 113 ; 2) 15 ; 36 ; 39 ; 3) 15 ; 20$; $25 ; 4) 15 ; 8 ; 17$.

Answer: 4 triangles.