Question

Aflati:

k apartine Z pentru care 16/2k+1 apartine Z


n aparine Z pentru care 6/n+2 apartine Z


n apartine N pentru care 8/3-n apartine Z

Answer 1

k apartine Z pentru care 16/2k+1 apartine Z

$\it \dfrac{16}{2}k+1 \in \mathbb{Z} \Longleftrightarrow 8k+1 \in \mathbb{Z} \Longleftrightarrow k \in \mathbb{Z}$
__________________________________________________________

n aparine Z pentru care 6/n+2 apartine Z


[tex]\it \dfrac{6}{n} +2 \in \mathbb{Z} \Longleftrightarrow \dfrac{6}{n} \in \mathbb{Z} \Longleftrightarrow n|6 \Longleftrightarrow n\in D_6 \Longleftrightarrow \\\;\\ \\\;\\ \Longleftrightarrow n \in \{\pm1,\ \pm2,\ \pm3,\ \pm6} \} [/tex]
_________________________________________________________


n apartine N pentru care apartine Z

n∈ N,  8/3∉ N ⇒8/3-n ∉ Z

Answer 2

1) (2k+1)∈D16     2k+1 ∈ {-16; -8; -4; -2; -1; 1; 2; 4; 8;16 }
2k ∈ {-17; -9; -5; -3; -2; 0; 1; 3; 7; 15} ⇒   k ∈{-1,0}
2)  (n+2) ∈D6     n+2 ∈ {-6; -3; -2; -1; 1; 2; 3; 6}   n∈{-8; -5; -4; -3; -1; 0; 1; 4}
3) (3-n) ∈ D8   3-n ∈ (-8; -4; -2; -1; 1; 2; 4; 8}  n ∈ {11;7;5;4;2;1;-1;-5}