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Suppose: $$a=(5^4 \pmod 7)^3 \pmod {13}$$

How would you go about solving this? The only thing I came up with is an ugly polynomial: $a=13k+7l^3+5^{12}$ but this doesn't tell me much. Any starting point would be great.

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    $\begingroup$ hint: $(5^4 \pmod 7)=2$ $\endgroup$ – Salech Rubenstein Dec 6 '13 at 3:22
  • $\begingroup$ gosh you're right $\endgroup$ – Dimitri Dec 6 '13 at 3:25
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$$5^2\mod 7 = -3$$ $$5^4 \mod 7 = 2$$. $$2^3=8$$.

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