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What is the remainder of the division: $$3^{302}\div{28}$$

Problem from a proofs book. I prefer hints. My goal is not to get homework done. I don't even attend a proofs class nor a number theory class; I do this for fun but this exercise was taking me too long; I will keep trying though, that is why I would like hints.

I have tried using Fermat's little theorem and modular cancellation theorem, since that is the theory background the book provides, but I haven't been able to figure it out yet.

Thanks for the quick answers!

O M
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Pallas cat
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    Hint: $27$ is a power of $3$ and happens to be very close to $28$ – JMoravitz Dec 29 '23 at 04:42
  • Pleas eprovided mroe ocntext. Where does the question come from? What have you you tried? Please add more details in your post via edit, as this kind of posts are highly discouraged and may face downvotes and/or closure. – Divide1918 Dec 29 '23 at 04:43
  • How are you able to use stackexchange but no calculator? – morrowmh Dec 29 '23 at 04:49
  • You can legally use negative numbers with mod math try out a small numbers to convince yourself it works – RandomGuy Dec 29 '23 at 05:08
  • Apply linked mod order reduction, using $,3^6\equiv 1\pmod{!28},$ (being true both $!\bmod 4\ &\ 7)\ \ $ – Bill Dubuque Dec 29 '23 at 07:21

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$$ 3^3\equiv -1 \bmod(28) \\ 3^{302}\equiv 9(27)^{100} \equiv 9(-1)^{100} \bmod(28) $$

Edit: The great thing about a number being mod 1 or -1 in any base is that we can raise it to any power and its still one since we got the nice result that $3^3$ is 1 mod(28) we can now write 302 as 3(100)+2 and the entire $3^{300}\equiv1$ which leaves only 9 which is now the remainder

RandomGuy
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