Question

E=(x/x apartine N si x+7 divizibil cu x+1)
aflati E. Dau coronita​

Answer 1

Explicație pas cu pas:

$\dfrac{x + 7}{x + 1} = \dfrac{x + 1 + 6}{x + 1} = 1 + \dfrac{6}{x + 1} \\ \implies (x + 1) \in \mathcal{D}_{6} \iff (x + 1) \in \Big\{ 1; 2; 3; 6\Big\} \\ \iff x \in \Big\{ 0; 1; 2; 5\Big\}$

$\implies E = \Big\{ 0; 1; 2; 5\Big\}$

Answer 2

$x + 7 \:\vdots \: x + 1 \\ \\ \implies \frac{x + 7}{x + 1} \in\mathbb{N} \\ \\ \frac{x + 1 + 6}{x + 1} = 1 + \frac{6}{x + 1} \\ \\ \\ \\ \\ \\ \\ \\ \\ \implies \: 1 + \frac{6}{x + 1} \in\mathcal{D}_{6} \\ \\ \implies \frac{6}{x + 1} \in\mathcal{D} _{6} \\ \\ \implies \frac{6}{x + 1} \in\{\pm1,\pm2,\pm3,\pm6\} \\ \\ \implies x + 1\in\{ \frac{6}{ - 1},\frac{6}{1} , \frac{6}{ -2},\frac{6}{ 2},\frac{6}{ -3},\frac{6}{ 3},\frac{6}{ -6},\frac{6}{ 6}\} \\ \\ \implies x + 1\in\{ - 6,6, - 3,3, - 2,2,- 1,1\} \\ \\ \implies x\in\{ - 7,5,- 4,2,- 3,1 - 2,0\}\\ \\ \iff x\in\mathbb{N}\\ \\\implies \boxed{x\in\{5,2,1,0\} }$