SMS1103: TUTORIAL 9
Multiplicative Functions, Euler phi Function, Eulers theorem
Question 1. [] Determine whether each of the following arithmetic function is completely multiplicative. Prove your answer.
(a) f (n) = 0
(b) f (n) = log n
(c) f (n) = n + 1

SMS1103: TUTORIAL 4
Question 1. Find all positive integers with exactly 2 dierent prime factors with all prime factors less than
10. Then nd all positive integers with exactly 4 dierent prime factors with all prime factors less than 15.
Question 2. Find a

SMS1103: TUTORIAL 8
Wilsons Theorem and Fermats Little Theorem
Question 1. [] Find inverses modulo 11 each of the following integers.
(a) 2
(b) 5
(c) 3
(d) 9
Question 2. Group together pairs of inverses modulo 7 that occur in 6! = 6 5 4 3 2 1.
Question 3.

SMS1103: TUTORIAL 6
Linear Congruences
Question 1. [] a 2 (mod 5) is the class of cfw_., 3, 2, 7, 12, 17.. Based on the same class, express this
class using modulo 15.
Question 2. [] Write the following linear congruences as linear diophantine equation.
(

SMS1103: TUTORIAL 7
System of Linear Congruences
Question 1. [] Find solutions of each of the following system of linear equations.
x+y
=2
x + 3y
(a)
=1
(b)
2x + y
=3
2x + y
=1
4x + 2y
=2
(c)
2x + 6y
=3
[In which cases, there is unique solution, no soluti

SMS1103: TUTORIAL 10
Character Ciphers
Question 1. Use the Caesar cipher, encrypt the message ATTACK AT DAWN.
Question 2. Decrypt the ciphertext message LFDPH LVDZL FRQTX HUHG which has been encrypted using
the Caesar cipher.
Question 3. Encrypt the messa

SMS1103: TUTORIAL 2
Divisibility
Question 1. [] Show that 3|99, 5|145, 7|343, and 888|0.
Question 2. [] Show that 1001 is divisible by 7, by 11, and by 13.
Question 3. [] Decide integers are divisible by 7?
(a) 0
(b) 123321
(d) 285714
(c) 707
Question 4.

SMS1103: TUTORIAL 1
CHAPTER ZERO:PRELIMINARIES
Sequence
Question 1. Find the sum of a geometric progression: 2, 4, 8, ., 128.
Question 2. Conjecture a formula for the nth term of cfw_an , if the first ten terms of this sequence are
as follows:
(a) 3, 11,

SMS1103: TUTORIAL 3
Greatest Common Divisors
Question 1. Find the greatest common divisor for each of the following pairs of integers
(a) 15, 35
(b) 11, 121
(c) 99, 100
(d) 0, 111
(e) 12, 18
(f) 100, 102
(g) 143, 227
(h) 306, 657
Question 2. For each ques