4. How many different right-angled triangles exist, one of the legs of which is equal to $\sqrt{2016}$, and the other leg and the hypotenuse are expressed as natural numbers

ANSWER: 12.
Solution. According to the condition $c^{2}-b^{2}=a^{2}=2016$, that is, $(c-b)(c+b)=2^{5} \cdot 3^{2} \cdot 7$. The system $\left\{\begin{array}{l}c-b=n, \\ c+b=k\end{array}\right.$ (here $n-$ is one of the divisors of the number 2016, and $k=\frac{2016}{n}$) has natural solutions $c=\frac{n+k}{2}, b=\frac{k-n}{2}$, if $n<k$ (that is, $n \leq 44$) and $n$ and $k$ are even.

Possible values of $n$: $2,2^{2}=4,2^{3}=8,2^{4}=16,2 \cdot 3=6,2^{2} \cdot 3=12,2^{3} \cdot 3=24$, $2 \cdot 7=14,2^{2} \cdot 7=28,2 \cdot 3^{2}=18,2^{2} \cdot 3^{2}=36,2 \cdot 3 \cdot 7=21$ - a total of 12 options.