TITLE: Calculate $11^{35} \pmod{71}$
QUESTION [6 upvotes]: Calculate $11^{35} \pmod{71}$
I have:
$= (11^5)^7 \pmod{71}$
$=23^7 \pmod{71}$
And I'm not really sure what to do from this point..

REPLY [2 votes]: If you know that $11^{4} \equiv 15 \bmod 71$ (taken from user7530 above), then:
$$ 11^{35} \equiv (-60)^{35} \equiv -(4 \cdot 15)^{35} \equiv -(2^2 \cdot 11^4)^{35} \equiv -(2 \cdot 11^2)^{70} \equiv -1 \bmod 71 $$