Question

Calculati restul impartirii numarului 123^457 la 7

Answer 1

$\displaystyle 123 \equiv 4 \pmod 7 \\ \\ 123^2 \equiv 4^2 \pmod 7 \equiv 2 \pmod 7 \\ \\ 123^3 \equiv 2 \cdot 4 \pmod 7 \equiv 1 \pmod 7 \\ \\ Rezulta ~123^{3n} \equiv 1 \pmod 7~\forall~n \in\mathbb{N}. \\ \\ In~particular~123^{456} \equiv 1 \pmod 7. \\ \\ 123^{457}=123^{456} \cdot 123 \equiv 1 \cdot 4 \pmod 7 \equiv 4 \pmod 7. \\ \\ Restul~este~4.$