I2.1 If the solution of the system of equations $\left\{\begin{array}{r}x+y=P \\ 3 x+5 y=13\end{array}\right.$ are positive integers, find the value of $P$.
$5(1)-(2): 2 x=5 P-13 \Rightarrow x=\frac{5 P-13}{2}$
(2) $-3(1): 2 y=13-3 P \Rightarrow y=\frac{13-3 P}{2}$
$\because x$ and $y$ are positive integers $\therefore \frac{5 P-13}{2}>0$ and $\frac{13-3 P}{2}>0$ and $P$ is odd $\frac{13}{5}<P<\frac{13}{3}$ and $P$ is odd $\Rightarrow P=3$