[tex]\displaystyle\\
\frac{3}{p+1}~~\text{se simplifica, dar numaratorul = 3 care este numar prim.}\\\\
\Longrightarrow~~\frac{3}{p+1}~~\text{se simplifica cu 3.}\\\\
\Longrightarrow~~\boxed{(p+1)~\vdots~3}\\\\
p \ \textgreater \ 2~~\text{ si p este numar natural prim.}\\
\Longrightarrow~~\text{p este numar impar.}\\
\Longrightarrow~~\text{(p + 1)este numar par.}\\\\
\Longrightarrow~~\boxed{(p+1)~\vdots~2 }\\\\
\text{Din: }~~(p+1)~\vdots~3~~\text{si}~~(p+1)~\vdots~2~~ \Longrightarrow~~\boxed{\boxed{(p+1)~\vdots~6}}[/tex]
[tex]\displaystyle\\
\frac{p+7}{6} = \frac{p+1 + 6}{6}\\\\
(p+1)~\vdots~6~~\text{si}~~ 6 ~\vdots~6\\
\Longrightarrow~~\text{Suma lor este divizibila cu 6.}\\
\Longrightarrow~~[(p+1) + 6]~\vdots~6\\
\Longrightarrow~~(p+7)~\vdots~6\\\\
\Longrightarrow~~ \frac{p+7}{6}~~\text{se poate simplifica cu 6.}\\\\
\Longrightarrow~~ \boxed{\boxed{\frac{p+7}{6}\in \mathbb{N}}}
[/tex]
