SMA 2203 NUMBER THEORY
UNIVERSITY EXAMINATION 2013/2014
FIRST SEMESTER EXAMINATION FOR THE DEGREE OF BACHELOR OF
SCIENCE IN MATHEMATICS AND COMPUTER SCIENCE
SMA 2203: NUMBER THEORY
th
Time: 9.00am – 11.00am
Date: Friday, 14 February 2014
INSTRUCTIONS
ANSWER QUESTION ONE AND ANY OTHER TWO QUESTIONS
Question One (30mks)
(g) Let d: ℝ → ℝ be a function defined as d(x,y) = x – y. Waitherero claims that d is an
2
equivalence relation. Show if she is correct.
(3mks)
(h) State without proof the Fermat’s little theorem.
(2mks)
(i) Give a unit of the form ab 2 where a 

0,b 0 in Q( 2) . Hence show why it is a
unit
(3mks)
(j) State the division algorithm. Hence, find quotient and remainder when (-30001) is
divided by 301.
(4mks)
(k) Express (1010101,50505) as a linear combination of 1010101 and 50505
(3mks)
(l) Use the Eratosthenes sieve to find the primes between 1 and 100.
(3mks)
(m)Prove that n\a if and only if a 0(mod n)
(3mks)
(n) Show that any positive integer can be written as a product of primes
(5mks)
(o) Define the following terms:
i.
Units
ii.
Trace
(2mks)
Question Two (20mks)
(a) Show that there are no other three consecutive odd integers except 3,5,7 such
that they are
all primes
(8mks)
(b) Prove that there are infinitely many prime integers
(8mks)
(c) Find ten consecutive composites.
(4mks)
Question Three (20mks)
(a) Prove that
(5mks)
3 is an irrational number
(b) Show that the congruence relation a b (mod m) where m is a positive integer
is an
equivalence relation
(5mks)
(c) Let a, b, c, d, r be integers. Show that if a b (mod n) and c d (mod n), then
i.
ac  bd (mod n)
(3mks)
ii.
rc  rb (mod n)
(3mks)
99
(d)
find 99 module 13
(4mks)
Question Four (20mks)
(a) Determine the continued fraction of
7.
7 to give the rational approximation of
(8mks)
(b) Determine which of the equations: 12x + 18y = 50 and 17x + 13y = 111, has
integer
solutions.
(4mks)
(c)
Determine
if x2 – 3y2 =  1 has integer solutions by doing computation
modulo
4. Hence
find any solution.
(8mks)