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Problem 17(Computing Powers Modn Conjecture)
We need to be able to compute $a^k\pmod n$.
- Compute $2^k\pmod 5$ for $k=1,2,3,4,5,6,7$. What pattern do you see. What is $2^k\pmod 5 $ for $k=257$ and for $k=49827512$.
- Compute $5^k\pmod {11}$ for $k=1,2,3,4,5,6,7,8,9,10,11,12$. Can you discover a pattern. What if $k=3047$ or $k=478209183234$?
- Make a conjecture about $a^k\pmod n$. Come up with a different example to back your conjecture.
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