Question

aratati ca fractia 7n +10
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5n+7 este ireductibila oricare ar fi n apartine N .(NR NAT)

Answer 1

$\text{Fie~d~nr.~cu~care~simplificam~astfel~incat:}~ \\ d~|~7n+10 \\ ~si~ \\ d~|~5n+7. \\ \\ d~|~7n+10/\cdot5\Rightarrow d~|~35n+50(1) \\ d~|~5n+7/\cdot7\Rightarrow d~|~35n+49(2)~\\ \\ Din~(1)~si~(2)\Rightarrow \text{d~divide~si~diferenta}\Rightarrow \\ d~|~35n+50-35n-49\Rightarrow d~|~1\Rightarrow \boxed{d=1}\Rightarrow \text{Fractia~este~ireductibila.}$