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intsu11h3

# intsu11h3 - Introductory Number Theory Homework 3 Due...

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Introductory Number Theory. Homework 3 Due: Tuesday, June 7, 2011, 4:45PM 1. Let m, n be integers, not both 0. Consider the set A = { mx + ny : x, y Z } . Show that this set coincides with the set of all multiples of the greatest common divisor of m, n . That is, if d = gcd ( m, n ), show A = { kd : k Z } . 2. If a, m N and gcd ( a, m ) = 1, then the order of an integer a modulo m is the smallest positive integer k such that a k 1 (mod m ). Prove: If gcd ( a, m ) = 1 and if k is the order of a modulo m , then k | φ ( m ). Hint: Divide φ ( m ) by k ; what can you say about the remainder? 3. Assume p 3 is prime and that p 6 | a and a 6≡ 1 (mod p ). Prove: p - 2 X k =1 a k = a + a 2 + · + a p - 2 ≡ - 1 (mod p ) . (of course, you do not have to prove the equality; that’s just the definition of the summation expression; you only have to prove the congruence.) 4. Textbook, Exercise 13.3 (p. 89). 5. Textbook, Exercise 14.1 (p. 94). 6. Determine the highest power of 3 dividing 1000!. 7. A nice formula involving Euler’s φ function is the following: Let n N and let d 1 , . . . , d r be all the positive divisors of

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intsu11h3 - Introductory Number Theory Homework 3 Due...

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