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USPshnik [31]
3 years ago
8

Using fermat's little theorem, find the least positive residue of $2^{1000000}$ modulo 17.

Mathematics
1 answer:
torisob [31]3 years ago
6 0
Fermat's little theorem states that
a^p≡a mod p

If we divide both sides by a, then
a^{p-1}≡1 mod p
=>
a^{17-1}≡1 mod 17
a^{16}≡1 mod 17

Rewrite
a^{1000000} mod 17  as
=(a^{16})^{62500} mod 17
and apply Fermat's little theorem
=(1)^{62500} mod 17
=>
=(1) mod 17

So we conclude that
a^{1000000}≡1 mod 17

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