Explicație pas cu pas:
[tex] \frac{2x - 3}{x + 5} = \frac{2x + 10 - 13}{x + 5} = \frac{2(x + 5) - 13}{x + 5} \\ = 2 - \frac{13}{x + 5} \in \mathbb{Z} \\ 2 \in \mathbb{Z} \implies \frac{13}{x + 5} \in \mathbb{Z} \\ (x + 5)\in \{-13;-1;1 ;13 \}[/tex]
[tex]x + 5 = - 13 \implies x = - 18 \\ x + 5 = - 1 \implies x = - 6 \\ x + 5 = 1 \implies x = - 4 \\ x + 5 = 13 \implies x = 8[/tex]
G = {-18; -6; -4; 8}
[tex]{x}^{2} + x = 2 \cdot (1 + 2 + ... + 2021) \\ x(x + 1) = 2 \cdot \frac{2021 \cdot 2022}{2} = 2021 \cdot 2022 \\ x \in \mathbb{N} \implies x = 2021[/tex]
H = {2021}
