<s>[INST] how to solve $x^{113}\equiv 2 \pmod{143}$ I need to solve $x^{113} \equiv 2 \pmod{143}$ 
$$143 = 13 \times 11$$
I know that it equals to $x^{113}\equiv 2 \pmod{13}$ and $x^{113}\equiv 2 \pmod{11}$
By Fermat I got
1)  $x^{5} \equiv 2 \pmod{13}$
2)  $x^{3} \equiv 2 \pmod{11}$
Now I'm stuck..
 [/INST] $x^5 \equiv 2 (\mod13)$
$x^{12} \equiv 1(\mod 13)$ (Femat's little theorem.)
$5\cdot5\equiv 1 (\mod 12)\\
(2^5)^5 \equiv 2 (\mod 13)\\
(2^5) \equiv 6 (\mod 13)\\
x\equiv 6 (\mod 13)$
Now use the same logic mod 11
$x^{10} \equiv 1(\mod 11)\\
3\cdot7 \equiv 1(\mod 10)\\
(2^7)^3 \equiv 2(\mod 11)\\
(2^7) \equiv 7(\mod 11)\\
7^3 \equiv 2(\mod 11)\\
x\equiv 7(\mod 11)$
Can you get home from here?
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