<s>[INST] Solve $4^{138x + 75} \equiv 4^{15} \pmod{53}$ I want to solve the equation
\begin{equation}
4^{138x + 75} \equiv 4^{15} \pmod{53}
\end{equation}
for $x$. Fermats little theorem yields $4^{52} \equiv 1 \pmod{53}$, but where do I go from here? 
 [/INST] $2^{13}=8192\equiv 30\pmod {53}$, then $2^{26}=900\equiv-1\pmod{53}$. Also, $2^4\not\equiv1\pmod{53}$. Therefore
$$2^n\equiv 1\pmod{53}\iff n\equiv 0\pmod{52}$$
Now,
$$2^{276x+150}\equiv 2^{30}\pmod{53}$$
$$276x+150\equiv 30\pmod {52}$$
$$69x\equiv -30\pmod {13}$$
$$23x\equiv -10\pmod{13}$$
Can you finish?
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