Spr10_test1_soln

Spr10_test1_soln - Name: Prasad Tetali Math 4150...

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Prasad Tetali Math 4150 – Introduction to Number Theory Spring 2010 Test # 1 You have 50 minutes to take this test. There are 52 total points possible. Simple calculators are ﬁne, but no programmable calculators are allowed. Please write your answers neatly and show all of your work. Feel free to write on both sides of each sheet. 1. (4+6=10 points) a. State Fermat’s Little Theorem. Soln . For p : prime and a : an arbitrary integer, a p a (mod p ). Or for p prime, and a such that ( a,p ) = 1, a p - 1 1 (mod p ). b. Let p and q be distinct primes. Let b be a positive integer so that GCD( b,pq ) = 1. Then explain why the following is true. b ( p - 1)( q - 1) 1 (mod pq) . Soln . One way to show this is to appeal to Euler’s theorem: for n and a positive integers so that ( a,n ) = 1, we have a φ ( n ) 1 (mod n ). Since φ ( pq ) = φ ( p ) φ ( q ) = ( p - 1)( q - 1), we are done. Alternately, we could use Fermat’s little theorem on primes p and q and note that b ( p - 1)( q - 1) ± b p - 1 ² q - 1 1( mod p ) , b ( p - 1)( q - 1) ± b q - 1 ² p - 1 1( mod

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This note was uploaded on 02/18/2010 for the course MATH 4150 taught by Professor Prasadtetali during the Spring '10 term at Georgia Tech.

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Spr10_test1_soln - Name: Prasad Tetali Math 4150...

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