Question

Determinati cate numere de forma 1x au proprietatea ca fractia ordinara 1/(1*x ) se transforma intr-o fractie zecimala:
a)finita;b)periodica simpla;c)periodica mixta

Answer 1

    
$Consider\;numitorul\;fractiei\; = \overline{1x} \\ P= \{x | x=numar~prim \} \\ A=\{2;~5\} \\ B=P-A \\ \\ \text{O fractie ordinara ireductibila se transforma in fractie zecimala:} \\ \bullet finita, \\ \text{daca toti factorii primi ai numitorului } \in A \\ \bullet periodica\;simpla, \\ \text{daca toti factorii primi ai numitorului } \in B \\ \bullet periodica\;mixta, \\ \text{daca avem factori primi al numitorului, care } \in A ~si ~ \in B$

$Rezolvare: \\ x \in \{0;~1;~2;~3;~4;~5;~6;~7;~;8~;9\} \\ \frac{1}{10} ~=\ \textgreater \ ~FZ-F~deoarece~10=2\cdot 5 ~unde~2~si~5 \in A \\ \frac{1}{11} ~=\ \textgreater \ ~FZ-PS~deoarece~11=11~unde~11 \in B \\ \frac{1}{12} ~=\ \textgreater \ ~FZ-PM~deoarece~12=2^2\cdot3 ~unde~2 \in A ~si~3 \in B \\ \frac{1}{13} ~=\ \textgreater \ ~FZ-PS~deoarece~13=13~unde~13 \in B \\ \frac{1}{14} ~=\ \textgreater \ ~FZ-PM~deoarece~14=2\cdot7 ~unde~2 \in A ~si~7 \in B$

$\frac{1}{15} ~=\ \textgreater \ ~FZ-PM~deoarece~15=3\cdot5 ~unde~3 \in B ~si~5 \in A \\ \frac{1}{16} ~=\ \textgreater \ ~FZ-F~deoarece~16=2^4 ~unde~2 \in A \\ \frac{1}{17} ~=\ \textgreater \ ~FZ-PS~deoarece~17=17 ~unde~17 \in B \\ \frac{1}{18} ~=\ \textgreater \ ~FZ-PM~deoarece~18=2\cdot 3^2 ~unde~2 \in A~si~3 \in B \\ \frac{1}{19} ~=\ \textgreater \ ~FZ-PS~deoarece~19=19 ~unde~19 \in B$