Question

Sa se determine multimile:
B egal [x apartine Z cu propietatea ca: x egal 6n+7/3n+1,n apartine Z [
Incercati sa-l scrieti matematic pe foaie ca sa-l rezolvati.(am pb cu tastatura)
App,trebuie sa fie rezolvare de cls a 9-a

Answer 1

B = {x∈ ℤ | x = (6n+7)/(3n+1), n∈ ℤ}

[tex]\it n\in \mathbb{Z} \ \ \ \ (*) \\ \\ \\ \dfrac{6n+7}{3n+1} \in\mathbb{Z} \Rightarrow 3n+1|6n+7 \ \ \ \ \ (1) \\ \\ \\ 3n+1|3n+1 \Rightarrow 3n+1|(3n+1)\cdot2 \Rightarrow 3n+1|6n+2\ \ \ \ (2)[/tex]


[tex]\it (1), (2) \Rightarrow 3n+1|6n+7-6n-2 \Rightarrow 3n+1|5 \Rightarrow3n+1\in D_5 \Rightarrow \\ \\ \\ \Rightarrow 3n+1\in \{\pm1,\ \pm5\} \Rightarrow 3n+1\in \{-5,\ -1,\ 1,\ 5\}|_{-1} \Rightarrow \\ \\ \\ 3n\in\{-6,\ -2,\ 0,\ 4\}|_{:3} \stackrel{(*)}{\Longrightarrow} n\in\{-2,\ 0\}[/tex]

[tex]\it n = -2 \Rightarrow x = \dfrac{6\cdot(-2)+7}{3\cdot(-2)+1} = \dfrac{-5}{-5} =1 \\ \\ \\ n = 0 \Rightarrow x = \dfrac{6\cdot(0)+7}{3\cdot(0)+1} = \dfrac{7}{1} =7 \\ \\ \\ B = \{1,\ 7\}[/tex]