Question

1. sa se arate ca fractia este reductibila:
n²+n/2n+6
2. sa se arate ca fractiile sunt ireductibile (n=numar natural):
a) n/3n+1
b) 7n+10/5n+7​

Answer 1

$\it 1.\ \ \dfrac{n^2+n}{2n+6}=\dfrac{n(n+1)}{2(n+3)}\\ \\ \\ 2|n(n+1\ \ \c{s}i\ \ 2|2(n+3)) \Rightarrow frac\c{\it t}ia\ dat\breve a\ este\ reductibil\breve a$

$\it 2.\ a)\ \dfrac{n}{3n+1}\\ \\ \\ Fie\ d\in\mathbb{N}^*\ astfel\ ca\ \begin{cases}\it d|n \Rightarrow d|3n\\ \\ \c{s}i\\ \\ \it d|{3n+1}\end{cases} \Rightarrow d|3n+1-3n \Rightarrow d|1 \Rightarrow d=1 \Rightarrow\\ \\ \\ \Rightarrow \dfrac{n}{3n+1}\ este\ frac\c{\it t}ie\ ireductibil\breve a$