Explain solution of this system of non-linear congruence equations So I have a system of non-linear congruence equations (i may be wrong in terms):
\begin{cases} x^3 \equiv 21\ (\textrm{mod}\ 23) \\ x^5 \equiv 17\ (\textrm{mod}\ 23) \end{cases}
Somewhere I've read that to solve this system one should:


*

*Find the solution of $3\cdot a + 5\cdot b = 1$ with Extended Euclidean Algorithm

*Use $a$ and $b$ values from previous step in the next formula: $x \equiv 21^a\times17^b\ (\textrm{mod}\ 23)$

*If $a$ or $b$ is negative, then calculate the modular inverse of $21$ or $17$ and use it in second formula with $-a$ or $-b$


And I don't understand why is it working. I've tried to perform some calculations to get the second formula but didn't succeeded. : (


Can you please explain me this?

Since neither $3$ or $5$ are divisors of $\varphi(23)=22$, both the maps
$$ f:x\mapsto x^3,\qquad g:x\mapsto x^5 $$
are bijective on $\mathbb{F}_{23}$. In particular, since $3^{-1}\equiv 15\pmod{22}$ and $5^{-1}\equiv 9\pmod{22}$,
$$ f^{-1}:x\mapsto x^{15},\qquad g^{-1}:x\mapsto x^9 $$
and $x^3\equiv 21\pmod{23}$ is equivalent to $x\equiv 7\pmod{23}$, as well as $x^5\equiv 17\pmod{23}$ is equivalent to $x\equiv 7\pmod{23}$. So $\color{red}{x\equiv 7\pmod{23}}$ is the wanted solution.
