4023f10ps2

4023f10ps2 - n; n + 10). 6. If m is an integer, then m Z =...

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Homework #2 Due: September 8, 2010 1. Find d = gcd(475 ; 385) and express it as a linear combination of 475 and 385. That is write d = 475 s + 385 t for some integers s and t . 2. (a) Calculate d = gcd(4307 ; 1121) and express it as a linear combination of 4307 and 1121. (b) Calculate m = lcm[4307 ; 1121]. Hint: You may want to refer to the formula on page 22 of the handout. 3. Which integers can be expressed in the form 12 m + 20 n , where m and n are integers? 4. Let a be an integer such that ( a; 72) = 6 and ( a; 245) = 7. Let b = 3 4 ¢ 5 3 ¢ 7 3 . (a) Compute ( a; b ). (b) What is the highest power of 7 which divides [ a; b ]? 5. If n ia a positive integer, flnd all of the possible values of gcd(
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Unformatted text preview: n; n + 10). 6. If m is an integer, then m Z = f mx : x 2 Z g is the set of all integer multiples of m . (a) If m and n are given integers, then give a precise description of the intersection m Z \ n Z : More precisely, m Z \ n Z = r Z for some integer r . Describe r in terms of m and n . (b) Similar to part (a), describe the set m Z + n Z = f mx + ny : x; y 2 Z g = s Z ; by describing the integer s . 7. Give a proof by induction to show that 5 2 n ¡ 1 is divisible by 24, for all positive integers n . Math 4023 1...
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This document was uploaded on 12/28/2011.

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