[tex]f:N \rightarrow N, f(n)= \text{restul impartirii numarului n la 5.} \\ a) \\ 0 : 5=0~rest~0~~\Longrightarrow~~f(0)=0 \\ 2 : 5=0~rest~2~~\Longrightarrow~~f(2)=2 \\ 3 : 5=0~rest~3~~\Longrightarrow~~f(3)=3 \\ 5 : 5=1~rest~0~~\Longrightarrow~~f(5)=0 \\ 7 : 5=1~rest~2~~\Longrightarrow~~f(7)=2 \\ \\[/tex]
[tex]b) \\ n=5k+r ~~~~~unde~~k \in N;~~~r \in \{0;~1;~2;~3;~4\} \\ n+5=5k+r+5 =5(k+1) + r~~~~unde~~k \in N;~~~r \in \{0;~1;~2;~3;~4\} \\ \\ n : 5 = (5k+r):5 =k ~rest~r~~\Longrightarrow~~f(n)=r \\ (n+5) : 5 = [5(k+1) + r]:5 = (k+1) ~rest~r~~\Longrightarrow~~f(n+5)=r \\ \Longrightarrow~~\boxed{f(n)=f(n+5)} ~~~\text{pentru orice } ~n \in N \\ cctd[/tex]
