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How can I find out what is the remainder when I divide $7^{220}$ by $8$ using modular arithmetic and without using any theorems such as Fermat's Little Theorem or Chinese Remainder Theorem?

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  • $\begingroup$ Any odd square is congruent to $1$ modulo $8$. $\endgroup$ Commented Nov 13, 2015 at 3:37

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Hint. $7\equiv -1 \pmod 8$, and $220$ is even ...

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    $\begingroup$ Thank you! 7 is -1 (mod 8), so 7^220 is (-1)^220 (mod 8), which means 7^220 is 1 (mod 8) and therefore the remainder is one! $\endgroup$
    – user265554
    Commented Sep 11, 2015 at 15:30
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    $\begingroup$ @user265554: Exactly! $\endgroup$ Commented Sep 11, 2015 at 15:33

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