Compute $5^{15} \pmod 7$.
Can someone help me understand how to solve this? I know there is a trick, but my professor did not completely explain it in class and I'm stuck.
Answer
You know 7 is prime and 7 does not divide 5 so you can use Fermats Little Theorm to get $5^6\equiv1 (mod 7)$ $\Rightarrow$ $5^{15} \equiv 5^3 (mod 7)$
then you can do $ (25)(5)\equiv (-4)(2) (mod7) $ then $ -8 \equiv 6 (mod 7)$ $\Rightarrow $ $5^{15} \equiv 6 (mod 7)$ hence $5^{15}$ Modulo 7 is 6
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