Assignments 04
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This document contains an assignment with 4 questions on topics related to number theory and modular arithmetic. Question 1 asks to prove several statements about greatest common divisors and divisibility. Question 2 involves computing greatest common divisors and finding solutions to linear congruences. Question 3 proves several statements about modular arithmetic. Question 4 asks to solve systems of congruences using the Chinese Remainder Theorem. Students are instructed to submit their answers by the due date to the course coordinator.
Assignments 04
- 1. 1 Academic Year: 2012/2013 Course Code: MPZ4140/MPZ4160 Assignment No. 04 Answer all questions (01). Show that (i). if m | ab, and (m,a )= 1, then m | b. (ii) gcd (a, b) = gcd (a, b-a) = gcd (a, b-2a) (iii) If 3 | 3x-y2 , then 3 | 3x2 – 3xy-xy2 +y3 +3x+12y (iv) Suppose a,b,n Z are such that n | a and n | b. Then n | gcd (a,b). (02). (i). Compute gcd ( 803, 154) and gcd (123, 147, 56). (ii). Find gcd (125, 962) and find integers x0, y0 that satisfy the linear equation 125x0+962y0 = 18. (iii). Find the general solution of the gcd (325, 4223) = 325x+4223y in integers x and y. (03). (i). Let a, b, c, d, n1, n2, m .Z (a) If )(mod 1nba and )(mod 2nca , then prove that )(modncb , where the integer n= gcd (n1, n2) . (b) If )(modmba and )(modmdc , then prove that )(modmacbd . (c) If )(modmba and )(modmdc , then prove that ))(mod2(2 mdbca . Department of Mathematics & Philosophy of Engineering Faculty of Engineering Technology The Open University of Sri Lanka Nawala - Nugegoda
- 2. 2 (04) Solve the following sets of congruence simultaneously (chinese Remainder theorem). (I) (mod2x 5) (mod5x 11) (mod7x 13) (mod3x 17) (II) (mod3x 5) (mod7x 9) (mod5x 17) (mod13x 19) Please send the answers to the following address on or before due date according to the activity schedule. Course Coordinator MPZ4140 or MPZ4160 Department of Mathematics & Philosophy of Engineering Faculty of Engineering Technology The Open University of Sri Lanka Nawala Nugegoda Virtual class Username: student0 Password: MPZ4140