Question

Aratati ca fractia 11n+ 3 supra 7n+ 2 este ireductibila,oricare ar fi n e N

Answer 1

$\rm Fie~d~un~numar~natural~nenul~astfel~incat~d~|~11n+3,~si~d~|~7n+2. \\ \\ d~|~11n+3 \Rightarrow ~d~|~7(11n+3) \Leftrightarrow d~|~77n+21. \\ \\ d~|~7n+2 \Rightarrow ~d~|~11(7n+2) \Leftrightarrow d~|~77n+22. \\ \\ d~|~77n+21~si~d~|~77n+22 \Rightarrow d~|~(77n+22)-(77n+21) \Leftrightarrow d~|~1. \\ \\ Insa~d \in N^* \Rightarrow d=1,~deci~numerele~11n+3~si~7n+2~sunt~prime~intre~ \\ \\ ele,~si,~deci, \frac{11n+3}{7n+2}~este~o~fractie~ireductibila.$