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Unformatted text preview: S UMMARY OF 9/29 L ECTURE In the previous lecture we proved that given any finite abelian group G and any a ∈ G , a  G  = 1 . We began this lecture with a corollary of this result: Theorem 1 (Euler’s Theorem) . a ϕ ( n ) ≡ 1 ( mod n ) for every integer a coprime to n . We then went over an application. Note that any integer a can be written in the form a = 10 q + r , where ≤ r ≤ 9 is the last digit of a ; in particular, we have a ≡ r ( mod 10) . Euler’s theorem, combined with this observation, allows us to determine the last digit of almost any number raised to a power. Indeed, we have ϕ (10) = 4 , so for any integer a coprime to 10, a 4 ≡ 1 ( mod 10) , and therefore must have last digit 1! This implies that a 8 also has last digit 1, etc. For example, the last digit of 3173 13 can be calculated as follows: 3173 13 = (3173 4 ) 3 · 3173 ≡ 1 3 · 3 ( mod 10) ≡ 3 ( mod 10) so the last digit is 3....
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 Fall '10
 GideonMaschler
 Ring, Cyclic group, shifted copies

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