Question

Aratati ca fractia 2n+7/n+3 este ireductibila,oricare ar fi n apartine N

Answer 1

$\displaystyle \frac{2n+7}{n+3} \\ \\ Fie~d=(2n+7,n+3)\\ \\ \left \{ {{d|(2n+7) \atop {d|(n+3) \Rightarrow d|2(n+3) \Rightarrow d|(2n+6)}} \right. \Rightarrow \\ \\ \Rightarrow d|[(2n+7)-(2n+6)]\Rightarrow d|(2n+7-2n-6) \Rightarrow d|(7-6) \Rightarrow d|1 \Rightarrow d=1 \\ \\ (2n+7,n+3)=1 \\ \\ \Rightarrow \frac{2n+7}{n+3} ~este~ireductibila$