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How to solve: ${9x^7\equiv67\pmod{149}}$

I've already found out that $x^7\equiv24\pmod{149}$ by using Extended Euclidean Algorithm But I have no clue what to do next. Does anyone have any idea?

Bill Dubuque
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Gordon Z
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    Hint: $149$ is prime, and $148$ is relatively prime to $7.$ – Thomas Andrews Apr 07 '23 at 15:53
  • Hint 2: maybe you can find a number $k$ such that $(x^7)^k\equiv x\pmod{149}$ for all $x$. Then you have that $24^k=x^{7k}=x$ and you have found $x$ :D – student91 Apr 07 '23 at 15:58
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    I think I got it: By Euler's theorem, I could know that $x^{t} ≡ x^{t:mod:Φ(149)}:(mod:149)$ , so if I let inverse of $7(mod:148)$ be k , I could get: $x^{7k} ≡ x^{7k:mod:Φ(149)} ≡ x^1:(mod:149) $ – Gordon Z Apr 07 '23 at 16:14
  • @GordonZ I updated the dupe link to one that is much more pertinent. – Bill Dubuque Apr 07 '23 at 18:10

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