Question

Determinati elementele urmatoarelor multimi:

c)C={x ∈ ℕ| 3x -2 pe x +4 ∈ ℕ}
d)D={x ∈ ℕ | 4x + 3 pe 5x - 1 ∈ ℕ}

Answer 1

[tex]\frac{3x-2}{x+4}=\frac{3x+12}{x+4}-\frac{14}{x+4}=2-\frac{14}{x+4}\in\mathbb{N}\Leftrightarrow \frac{14}{x+4}\in\{0,1,2 \}\\ \frac{14}{x+4}=1\Rightarrow x=10 \\ \frac{14}{x+4}=2\Rightarrow x=3 \\ C=\{10,3 \} \\ \\ \frac{4x+3}{5x-1}\in\mathbb{N}\Rightarrow 5x-1|4x+3\Rightarrow 5x-1|20x+15 \\ \text{Dar }5x-1|20x-4 \\ \text{Din ultimele doua relatii-scazand: }\\ 5x-1|19\Rightarrow 5x-1=1\text{ sau }5x-1=19 \\ \text{Rezulta $x=5$ este singura solutie convenabila} \\ D=\{4\}[/tex]