[tex]\displaystyle \\ \texttt{Suntem in multimea numerelor naturale} \\ \\
Sn = \frac{(2x+29)\cdot n}{2} \\
Sn=165 \\ \\
\frac{(2x+29)\cdot n}{2} = 165 \\ \\
(2x+29)\cdot n = 165 \cdot 2 \\
(2x+29)\cdot n = 330 \\
\texttt{Situatia: Un produs de 2 factori necunoscuti = 330}
[/tex]
[tex]\displaystyle \\
F1 = (2x+29) \\
F2 = n \\
\texttt{Il descompunem pe 330. } \\
330 = 2 \cdot 3 \cdot 5\cdot 11 \\
\texttt{Divizorii lui 330 sunt: } \\
D_{330} = \{ 1; 2; 3; 5; 6; 10; 11; 15; 22; 30; 33; 55; 66; 110; 165; 330 \} \\ \\
(2x+29) \cdot n= 330 \\ \\
n = \frac{330}{(2x+29)}
[/tex]
[tex]\displaystyle \\
(2x+29) \ \textgreater \ 29 \\ \Longrightarrow~~(2x+29) \text{poate fi egal cu oricare divizor al lui 330 } \\ \text{mai mare decat 29, cu conditia ca x sa fie numar intreg} \\
2x+29 = 30 ~~ \Longrightarrow~~ x = \frac{30-29}{2} = \frac{1}{2} \notin N ~~~~\text{solutie respinsa} \\ \\
2n + 29 = 33 ~~ \Longrightarrow~~ x_1 = \frac{33-29}{2} = 2 ~~ \Longrightarrow~~ n_1 = \frac{330}{33} = 10
[/tex]
[tex]\displaystyle \\
2n + 29 = 55~~ \Longrightarrow~~ x_2 = \frac{55-29}{2} = 13 ~~ \Longrightarrow~~ n_2= \frac{330}{55} = 6 \\ \\
2n + 29 = 165~~ \Longrightarrow~~ x_3 = \frac{165-29}{2} = 68 ~~ \Longrightarrow~~ n_3= \frac{330}{165} = 2 \\ \\
\text{Nu am folosit divizorii 66, 110 si 330 deoarece sunt numere pare} \\
\text{si ar fi rezultat o valoare zecimala pentru }x.[/tex]
