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) (mod2x 5)
(mod5x 11)
(mod7x 13)
(mod3x 17)
(II) (mod3x 5)
(mod7x 9)
(mod5x 17)
(mod13x 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