Euler's Theorem to solve for X $x^{138} \bmod 77 = 25$. How can I use Euler's Theorem to solve for $x$. $77$ is not a prime number but its factors are. $7$ and $11$ are prime, so the totient function of $77$ will be $60$

$\displaystyle x^{138}\equiv25\pmod{77}\implies x^{138}\equiv25\pmod7\equiv4$
Clearly, $(x,7)=1$ so using Fermat's Little Theorem, $\displaystyle x^6\equiv1\pmod7$
As $138\equiv0\pmod6,x^{138}\equiv1\pmod7$
or $\displaystyle\implies x^{138}=(x^6)^{23}\equiv1^{23}\pmod7\equiv1$
So, we need $\displaystyle1\equiv25\pmod7\iff3\equiv0\pmod7\iff7|3$ which is impossible
Hence, no solution
Had there be any solution, we could simply for $\pmod{11}$ and finally apply CRT
