Strategies to solve congruence problems Which strategy is best to use when solving problems of the following sort?   $$x^{29} \equiv 3\pmod {184}$$ 

$x^{29}\equiv3\pmod {184}$
$\implies x^{29}\equiv3\pmod {23}$ as $184=23\cdot8$
Now, $\phi(23)=22\implies x^{22}\equiv1\pmod{23}\implies x^{29}\equiv x^7\cdot x^{22}\pmod{23}\equiv x^7$
$\implies x^7\equiv3\pmod{23}\implies x^{21}\equiv 3^3\pmod {23}\equiv 4$
So, $4\cdot x^{22}\equiv x^{21}\cdot1\pmod{23}\implies 4x\equiv1\pmod {23}\equiv 24$
So, $x\equiv6\pmod {23}$
Similarly, $ x^{29}\equiv3\pmod 8$ as $184=23\cdot8$
As $\lambda(8)=2, x^2\equiv1\pmod 8$
$\implies x^{29}=(x^2)^{14}\cdot x\equiv x\pmod 8\implies x\equiv3\pmod 8$
Now, apply Chinese remainder theorem, to find $x\equiv 75\pmod {184}$
