Question

1) Determinati valorile intregi ale lui x pentru care avem:

a) 10/x ∈ Z
b) -5/x-7 ∈ Z
c) 19/3x-2 ∈ Z
d) 2x+5/x-2 ∈ Z

Answer 1

$\it x\in\mathbb{Z}\ \ \ (*)$

$\it \dfrac{19}{3x-2} \in\mathbb{Z} \Rightarrow 3x-2|19 \Rightarrow 3x-2\in D_{19} \Rightarrow 3x-2\in \{\pm1,\pm19\}$

$\it 3x-2 \in \{-19, -1, 1, 19\}_{+2}\Rightarrow3x \in \{-17, 1, 3, 21\}_{:3} \stackrel{(*)}{\Longrightarrow} x\in\{1, 7\}$