solve the congruence equations

Question

solve the equations :

(1) $x^{11}\equiv 13$ mod $35$

(2) $x^5\equiv 3 $ mod $64$

how do we solve this problem can we little Fermat's theorem

Answer 1

Hint $\ (x,35) = 1\,$ so $\,x^4\equiv 1\pmod 5,\,$ and $\,x^6\equiv 1\pmod{7}\,$ so $\,x^{12}\equiv 1\pmod{35}$

therefore $\ x^{-1} \equiv x^{11} \equiv 13\,\Rightarrow\, x\equiv \dfrac{1}{13}\equiv\dfrac{3}{39}\equiv \dfrac{-32}{4}\equiv -8\equiv 27$