4. Francesco and Andrea decide to consult the mathematical oracle to find out if they have any lucky pairs $(x, y)$ of numbers (real numbers). To determine the pair (or pairs) of lucky numbers, the oracle asks both Francesco and Andrea for their birth day $(g)$ and month $(m)$, after which it solves the system for each of them:

$$
\left\{\begin{array}{l}
13 x-y=181 \\
g x-m y=362
\end{array}\right.
$$

The oracle's response is that Andrea has no lucky pairs of numbers, while Francesco has infinitely many lucky pairs of numbers. Which of the following statements is correct?

(A) Francesco and Andrea were both born in spring

(B) Francesco and Andrea were both born in summer

(C) Francesco and Andrea were both born in autumn

(D) Francesco and Andrea were both born in winter

(E) Francesco and Andrea were born in different seasons.
4. The answer is (D). Let's examine Francesco's situation: for the system to admit infinite solutions, it is necessary that the second equation be equivalent to the first, meaning they differ at most by a multiplicative constant. The constant term of the second equation is $362=181 \times 2$, so in Francesco's case, the coefficients of the second equation are twice those of the first, i.e., $g=26$ and $m=2$. Francesco was born on February 26.

In Andrea's case, however, the two equations must be inconsistent, which happens when the coefficients of the unknowns are proportional to each other, but the proportionality factor is not the same as that between the respective constant terms. This means that for Andrea, $g=13 \mathrm{~m}$. Given that $g \leq 31$, there are two possible values for $m$: 1 and 2. For $m=2$, however, the system has infinite solutions, so Andrea was born on January 13.

It is also possible to approach the problem geometrically by interpreting the equations as lines in the Cartesian plane and the solutions of the system as their intersections. In Francesco's case, the two equations must represent the same line, while in Andrea's case, they must represent two parallel but distinct lines. The conclusions are, of course, the same.