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I was working on a problem, I arrived at the point at which I have to find $17^{{{17}^{17}}^{17}} \pmod {25}$

My attempt: $$ 17^{{{17}^{17}}^{17}}\equiv 17^{{{{17}^{17}}^{17}} \pmod{\phi(25)}} \pmod {25} $$$$17^{{{17}^{17}}}\equiv 17^{{{{17}^{17}}} \pmod{\phi(20)}} \pmod{20}$$$$17^{17}\equiv1^{17}\equiv1\pmod{\phi(20)=8} $$ Thus:$$17^{{{17}^{17}}^{17}}\equiv 17^{17}\equiv17^{-8}\equiv(17^{-1})^{8}\pmod{25}$$ The inverse of $17$ modulo $25$ its $3$ since $17\cdot3=51$, so:$$17^{{{17}^{17}}^{17}}\equiv3^{8} \equiv3^3\cdot3^3\cdot3^2\equiv2\cdot2\cdot9\equiv36\equiv11\pmod{25}$$ I checked but the solutions says that it is actually congruent to $2$ and not $11$, everything before the inverse it's fine for sure since OP follows the same procedure, what did I do wrong?

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There is a mistake. $$17^{{{17}^{17}}^{17}}\equiv 17^{17}\equiv17^{\color{red}{-8}}\equiv(17^{-1})^{\color{red}{8}}\pmod{25}$$ should be $$17^{{{17}^{17}}^{17}}\equiv 17^{17}\equiv17^{\color{red}{-3}}\equiv(17^{-1})^{\color{Red}{3}}\pmod{25}$$

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    $\begingroup$ Wait, I know this might be a dumb question but... $17^{17} \neq 17^{17 \pmod {25}} \pmod{25}$ that's the error right? $\endgroup$ – Spasoje Durovic Dec 9 '18 at 12:25
  • $\begingroup$ How silly of me!... I had used the same logic more than once, then I completely messed it up at the end, Thanks! $\endgroup$ – Spasoje Durovic Dec 9 '18 at 12:30
  • $\begingroup$ Haha happens sometimes :) I don't see how CRT applies here though, you've just used lil fermat $\endgroup$ – AgentS Dec 9 '18 at 12:32
  • $\begingroup$ Yeah, CRT applied in a different part of the problem since i was evaluating that ugly $17^{17^{17^{17}}}$ thing in $\pmod{100}$ so I broke it down into two congruences, one modulo $4$ and the other modulo $25$ $\endgroup$ – Spasoje Durovic Dec 9 '18 at 14:26

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