3. Is it possible to represent the number 2017 as the sum of three natural numbers such that no two of them are coprime?
3. It is possible. Let's look for the required representation in the form, for example, $2017=6 x+10 y+15 z$ (then for any $x, y$, and $z$, the first two terms have a common divisor of 2, the first and third terms have a common divisor of 3, and the second and third terms have a common divisor of 5, so no two are coprime). It is sufficient to find one solution to this equation, which can be done in many ways: for example, by trying small values for $x$ and $y$ until we find a case where the expression $2017-6 x-10 y$ is divisible by 15 (and then we can take $z$ to be the quotient of those two numbers) we quickly find that this happens (one possibility) for $x=2, y=1$, and we calculate $z=\frac{2017-6 \cdot 2-10 \cdot 1}{15}=\frac{1995}{15}=133$. Thus, we have the representation $2017=12+10+1995$.