Please advise on a short cut on how to solve $$7^{3216645} \pmod {17}$$ by hand.
So far I thought of using method of squaring and fast exponentiation (converting to binary form...) but these methods seem to be long.
Thank you!
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Just use Fermat. $7^{16}\equiv 1 \pmod {17}$. – lulu Oct 26 '19 at 18:12
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and it's easy to see the exponent is 5 mod 16 @lulu – Oct 26 '19 at 18:39
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You can actually turn the exponent into a work around using mod 17, but it's a bit of a pain.. – Oct 26 '19 at 20:12
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in this case, you can use $$3216645=17\cdot 189214+7$$$$189214=17\cdot 11130+4$$$$11130=17\cdot 654+12$$$$654=17\cdot 38+8$$$$38=17\cdot 2+4$$ with a bit more of theory, to conclude it's congruent to $7^{37}$ which by the same logic is congruent to $7^5$ – Oct 29 '19 at 00:01