Question

Sa se determine toate elementele multimii: {x∈Z, $\frac{x+4}{x+1}$ ∈Z}

Answer 1

[(x+1)+3]/(x+1) =(x+1)/(x+1) +3/(x+1)=
1+3/(x+1)
trebuie sa determinam divizorii lui 3
D3={-3,-1,1,3}
x+1=-3
x=-4
x+1=-1
x=-2
x+1=1
x=0
x+1=3
x=2
S={-4,-2,0,2}

Answer 2

Notam A={x∈Z ,x+4/x+1 ∈Z}
x+4/x+1 ∈Z⇒x+1/x+4
dar x+1/x+1 oricare ar fi x∈Z ⇒din ambele relatii ca x+1/x+4-x-1
x+1/3⇒x+1∈D3(multimea divizorilor intregi ai lui 3);
x+1={-3;-1;1;3}
x={-4;-2;0;2};
A={-4;-2;0;2};