[tex]\it \dfrac{7x-5}{2x+1} \in \mathbb{Z},\ \ x\in \mathbb{Z}, \ \ \ x=?
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\dfrac{7x-5}{2x+1} \in \mathbb{Z} \Rightarrow 2x+1|7x-5 \Rightarrow 2x+1|2(7x-5) \Rightarrow
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\Rightarrow 2x+1|14x-10 \ \ \ (1)[/tex]
[tex]\it Dar,\ \ 2x+1|2x+1 \Rightarrow 2x+1|7(2x+1) \Rightarrow 2x+1|14x+7 \ \ \ (2)
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(1),\ (2) \Rightarrow 2x+1| 14x+7-(14x-10) \Rightarrow 2x+1|14x+7-14x+10
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\Rightarrow 2x+1|17 \Rightarrow 2x+1 \in D_{17} \Rightarrow 2x+1 \in\{\pm1,\ \pm17\} \Rightarrow[/tex]
[tex]\it \Rightarrow 2x+1\in \{-17,\ -1,\ 1,\ 17\}|_{-1} \Rightarrow 2x\in \{-18,\ -2,\ 0,\ 16\}|_{:2} \Rightarrow
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\Rightarrow x\in\-\{-9,\ -2,\ 0,\ 8\}.[/tex]
