intsu11h3s - Introductory Number Theory. Homework 3 Due:...

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Introductory Number Theory. Homework 3 Due: Tuesday, June 7, 2011, 4:45PM Note: My words to work on your own seem to continue to be ignored. At least, don’t make it so obvious! At least, UNDERSTAND WHAT YOU ARE COPYING!!! I know you are copying without really understanding when I see meaningless statements, or incomplete statements, that make sense when one sees the complete form in someone else’s paper. I may not fail you for it, but rest assured that you I am aware of it. My apologies to the few of you who work on their own and don’t allow others to copy from them. 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 } . Solution. By Bezout, there exist integers a,b such that d = am + bn . For every k Z we have kd = kam + kbn A , proving that the set { kd : k Z } ⊂ A . But one also has to prove the converse inclusion! Assume now x,y Z . Since d | m and d | n , there exists integers j,‘ such that m = jd , n = ‘d , hence mx + ny = jdx + ‘dy = ( jx + ‘y ) d = kd with k = jx + ‘y . The converse inclusion follows. 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? Solution. Since a φ ( m ) 1 (mod m ) by Euler’s theorem, we must have 1 k φ ( m ). By the division algorithm, we can write φ ( m ) = qk + r for unique integers q,r , 0 r < k . Now, once again by Euler’s theorem, and because a k 1 (mod m ), 1 a φ ( m ) = a qk a r = ( a k ) q a r 1 q a r = a r (mod m ); i.e., a r 1 (mod m ). If r > 0 then 1 r < k , contradicting that k is the least positive integer such that
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This note was uploaded on 07/13/2011 for the course MAS 3202 taught by Professor Staff during the Summer '11 term at FAU.

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intsu11h3s - Introductory Number Theory. Homework 3 Due:...

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