Sma2113:Maths For Science Question Paper
Sma2113:Maths For Science
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2013
QUESTION ONE - (30 MARKS)
a) Show that the set 2 × 2 matrices over integers form a non-commutative ring with unity under matrix addition and multiplication. (5 Marks)
b) Let be a non-zero integer. Show that if , then |. (4 Marks) c) Determine the least non-negative residue x such that 6100 = ( 17). (4 Marks) d) Show that if the HCF of two integers a and b exists, then it must be unique.(5 Marks) e) Compute the phi-function 36 . (4 Marks) f) Find the order of 3 modulo 13. (4 Marks) g) Prove that if ( ) then and have the same order. (4 Marks)
QUESTION TWO (20 MARKS)
a) Show that the = {0,1,2,3,4,5,6,7} under addition and multiplication modulo 8 is commutative but is not an integral domain. (6 Marks) b) Prove that if , and , = 1 then |. (5 Marks) c) Show that of | and |, then = ±. (4 Marks) d) Find all the solutions of the Diophantine equation 77 + 42 = 35. (5 Marks)
QUESTION THREE (20 MARKS)
a) i) Define a primitive root of a prime number P. (2 Marks)
2
ii) Show that 3 is a primitive root of 5 but 4 is not. (6 Marks) b) Determine whether 5 is a primitive root of 11. (5 Marks) c) Show that 28+5 = 2517 where m is any positive integer. (7 Marks)
QUESTION FOUR (20 MARKS)
a) Define the term ring as used in the set of integers. (6 Marks) b) Show that the set = {0,1,2,3,4,5,}is a ring under addition modulo 6. (6 Marks) c) i) Define the term co-prime as used in the set of integers. (2 Marks) ii) Prove that two integers a and b are co-prime if and only if there exist x,y such that + = 1. (6 Marks)
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