Question

19. Aflaţi x aparține N dacă a) (x+3)|(x+7); b) (x+1)|(2x+5); c) (x+2)| (3x+8); d) (2x+1)|(4x+9); e) (2x+3)|(3x +5)​

Answer 1

Explicație pas cu pas:

a) (x+3)|(x+7) <=> (x+3)|[(x+3)+4] => (x+3) | 4

(x+3) ∈ D(4) <=> (x+3) ∈ {1, 2, 4} => x ∈ {-2, -1, 1}

x ∈ N => x = 1

b) (x+1)|(2x+5) <=> (x+1)|[2(x+1)+3] => (x+1) | 3

(x+1) ∈ D(3) <=> (x+1) ∈ {1, 3} => x ∈ {0, 2}

c) (x+2)| (3x+8) <=> (x+2)|[3(x+2)+2] => (x+2) | 2

(x+2) ∈ D(2) <=> (x+2) ∈ {1, 2} => x ∈ {-1, 0}

x ∈ N => x = 0

d) (2x+1)|(4x+9) <=> (2x+1)|[2(2x+1)+7] => (2x+1) | 7

(2x+1) ∈ D(7) <=> (2x+1) ∈ {1, 7} => x ∈ {0, 3}

e) (2x+3)|(3x+5) <=>

(2x+3) | [2×(3x+5)] => (2x + 3) | (6x + 10)

(2x+3) | [3×(2x+3)] => (2x + 3) | (6x + 9)

=> (2x + 3) | [(6x + 10) - (6x + 9)]

=> (2x + 3) | 1

=> ((2x+3),(3x+5)) = 1 => sunt prime între ele

=> nu există x ∈ N a.î. (2x+3)|(3x+5)