modular exponentiation large power mod composite number $$x^{11} \equiv 12 \pmod{143}$$
How to solve this without any sophisticated methods?
$$x^{11} \equiv 12 \pmod{143}$$    
$${x^{120}}\equiv 12^{10}\pmod{143} $$
Is this correct knowing that x could divide 143?
Thanks in advance.

As $143=11\cdot13,$ $$x^{11}\equiv12\pmod{143}$$
$\implies(i)x^{11}\equiv12\pmod{11}\equiv1$
But by Fermat's little theorem, $x^{11-1}\equiv1\pmod{11}\implies x\equiv1\pmod{11}\ \ \ \ (1)$
and $(ii)x^{11}\equiv12\pmod{13}\equiv-1,$
Now $12x\equiv x^{12}\equiv1\pmod{13}$ by Fermat's little theorem
$\implies 1\equiv12x\equiv-x\iff \equiv-1\pmod{13}\ \ \ \ (2)$
Using Chinese remainder theorem/by observation, $$x\equiv12\pmod{\text{lcm}(11,13)}$$
