Question

Determination valorile numarului intreg x pentru care a) 10 pe x apartine N b) - 15 pe x apartine N c) 21 pe x+1 apartine Z d) 3 pe 7-2x apartine z

Answer 1

$\it a)\ \dfrac{10}{x}\in\mathbb{N} \Rightarrow x>0,\ x\in D_{10} \Rightarrow x \in \{1,\ 2,\ 5,\ 10\}\\ \\ \\ b)\ \dfrac{-15}{x}\in \mathbb{N} \Rightarrow x<0,\ x\in D_{15} \Rightarrow x\in\{-1,\ -3,\ -5,\ -15\}\\ \\ \\ \dfrac{21}{x+1}\in\matbb{Z} \Rightarrow x+1\in D_{21}\Rightarrow x+1\in\{\pm1,\ \pm3,\ \pm7,\ \pm21\} \Rightarrow \\ \\ \\ \Rightarrow x+1\in \{-21,\ -7,\ -3,\ -1.\ 1,\ 3,\ 7,\ 21\}|_{-1}\Rightarrow x\in\{-22,\ -8,\ -4,\ -2,\ 0,\ 2,\ 6,\ 20\}$

$\it \dfrac{3}{7-2x}\in\mathbb{Z} \Rightarrow 7-2x\in D_{3}\Rightarrow 2x-7\in D_3 \Rightarrow 2x-7\in \{\pm1,\ \pm3\}\Rightarrow \\ \\ \\ \Rightarrow 2x-7\in\{-3,\ -1,\ 1,\ 3\}|_{+7}\Rightarrow 2x\in\{4,\ 6,\ 8,\ 10\}|_{:2}\Rightarrow x\in\{2,\ 3,\ 4,\ 5\}$