Question

10. Arătaţi că numărul a = 7⁸³-3⁴¹se divide cu 10​

Answer 1

$u(7 {}^{x} ) = \begin{cases}7 \: \c{} \: daca \: x= 4k + 1 \\ 9 \: \c{} \: daca \: x = 4k + 2 \\ 3 \: \c{} \: daca \: x = 4k + 3 \\ 1 \: \c{} \: daca \: x = 4k\end{cases} \\ \\ u(3 {}^{x} ) = \begin{cases}\ 3 \: \c{} \: daca \:x =4k + 1 \\9 \: \c{} \: daca \: x = 4k + 2 \\ 7 \: \c{} \: daca \: x = 4k + 3 \\ 1 \: \c{} \: daca \: x = 4k \end{cases} \\ \\ u(7 {}^{83} ) = 3 \: \c{} \: deoarece \: \: 83 = 4 \times 20 + 3 \\ u(3 {}^{41} ) = 3 \: \c{} \: deoarece \: 41 = 4 \times 10 + 1 \\ u(7 {}^{83} - 3 {}^{41} ) = 3 - 3 = 0 \\ \iff \: daca \: \: se \: \: termina \: \: in \: \: 0 \\ \: \: \boxed{se \: \: imparte \: \: la \: \: 10}$