MATH135_F11_Assignment_7_Solutions

MATH135_F11_Assignment_7_Solutions - 1 MATH 135 F 2011...

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1 MATH 135 F 2011: Assignment 7 Due: 8:30 AM, Wednesday, 2011 Nov. 16 in the dropboxes outside MC 4066 Write your answers in the space provided. If you wish to typeset your solutions, use the solution template posted on the course web site. Typesetting is done in L A T E X. A very good online all-purpose L A T E Xmanual is the L A T E Xwikibook at http://en.wikibooks.org/wiki/LaTeX . Links to installations for various operating systems are also included there. Stephen Carr is IST’s LaTeX goto person, and his introductory guide is at http://www.ist.uwaterloo.ca/ew/saw/latex/latex_getstarted.pdf . L A T E Xis totally voluntary. Cite any proposition or definition you use. Family Name: Solutions First Name: I.D. Number: Section: Mark: (For the marker only.) If you used any references beyond the course text and lectures (such as other texts, discussions with colleagues or online resources), indicate this information in the space below. If you did not use any aids, state this in the space provided. 1. Provide the complete solution for each of the following. (a) x 3 (mod 7) x 5 (mod 12) Solution Since gcd(7 , 12) = 1, we know by the Chinese Remainder Theorem that a solution to this pair of linear congruences exists. Rewriting x 3 (mod 7) as x = 7 y + 3 (1) for y Z and substituting into the second linear congruence gives 7 y + 3 5 (mod 12). This reduces to 7 y 2 (mod 12) and the solution is y 2 (mod 12). Rewriting y 2 (mod 12) as y = 12 z + 2 for z Z and substituting in Equation 1 gives x = 7(12 z + 2) + 3 = 17 + 84 z for z Z , or, equivalently, x 17 (mod 84) Check 17 3 (mod 7) and 17 5 (mod 12). Also, 84 0 (mod 7) and 84 0 (mod 12).
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2 (b) x 7 (mod 10) x 5 (mod 12) Solution Since gcd(10 , 12) | (7 - 5), we know by Assignment 5, Question 1 that a solution to this pair of linear congruences exists. Rewriting x 7 (mod 10) as
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This note was uploaded on 12/31/2011 for the course MATH 135 taught by Professor Andrewchilds during the Spring '08 term at Waterloo.

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MATH135_F11_Assignment_7_Solutions - 1 MATH 135 F 2011...

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