Question

mostre que 13| 2 elevado a 70 + 3elevado a 70

Answer 1

Note que, $2^7=128\equiv11\equiv-2\pmod{13}$.

Assim, $2^{70}=(2^7)^{10}\equiv(-2)^{10}\equiv1024\equiv10\pmod{13}$.

Temos que, $3^3=27\equiv1\pmod{13}$.

Com isso, $3^{69}=(3^3)^{23}\equiv1^{23}\equiv1\pmod{13}$.

Portanto, $3^{70}=3\cdot3^{69}\equiv3\cdot1\equiv3\pmod{13}$.

E finalmente, $2^{70}+3^{70}\equiv10+3\equiv13\equiv0\pmod{13}$.