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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 (Eulers Theorem) . a ( 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) . Eulers theorem, combined with this observation, allows us to determine the last digit of almost any number raised to a power. Indeed, we have (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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This note was uploaded on 03/26/2011 for the course MAT 301 taught by Professor Gideonmaschler during the Fall '10 term at University of Toronto Toronto.
 Fall '10
 GideonMaschler

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