Question

Determina multimile E={x€N/27 pe 2x+1€N}
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Answer 1

$Pentru~ca~\frac{27}{2x+1} \in N~trebuie~ca~27~sa~fie~divizibil~cu~(2x+1).~Adica~(2x+1)~sa~apartina~multimii~divizorilor~naturali~ai~lui~27.\\\\ D_{27}=\{1,~3,~9,~27\}\\\\ Cazul~I:~2x+1=1~|-1 \Rightarrow 2x=0~|:2\Rightarrow x=0\\\\ Cazul~II:~2x+1=3~|-1 \Rightarrow 2x=2~|:2\Rightarrow x=1\\\\ Cazul~III:~2x+1=9~|-1 \Rightarrow 2x=8~|:2\Rightarrow x=4\\\\ Cazul~IV:~2x+1=27~|-1 \Rightarrow 2x=26~|:2\Rightarrow x=13\\\\ \Rightarrow x \in \{0,~1,~4,~13\}\\\\ x \in N \\\\ \Rightarrow E=\{0,~1,~4,~13\}$

Answer 2

$\it \dfrac{27}{2x+1} \in\mathbb{N} \Rightarrow 2x+1\in D_{27} \Rightarrow 2x+1 \in \{1,\ 3,\ 9,\ 27\}|_{-1}\Rightarrow \\ \\ \\ \Rightarrow 2x \in \{0,\ 2,\ 8,\ 26\}|_{:2} \Rightarrow x\in \{0,\ 1,\ 4,\ 13\}$