3. How many solutions in natural numbers does the equation

$$
(2 x+y)(2 y+x)=2017^{2017} ?
$$
Answer: 0.

Note that the sum of the numbers $A=2x+y$ and $B=2y+x$ is divisible by 3. Since the number on the right side is not divisible by 3, neither $A$ nor $B$ are divisible by 3. Therefore, one of these two numbers gives a remainder of 2 when divided by 3, and the other gives a remainder of 1. Thus, their product gives a remainder of 2. However, the number 2017 gives a remainder of 1, and therefore $2017^{2017}$ also gives a remainder of 1.

$\pm$ It is established that the system of equations cannot be solved in integers due to divisibility by 3, but there is no strict justification for why this will happen in all cases. 5 points

Ғ In the solution, several different factorizations are analyzed, and for each, it is established that the corresponding system of equations has no solutions. 3 points

- Correct answer without justification. 1 point