Proof by Contrapositive
July 12, 2012
So far weve practiced some different techniques for writing proofs. We started with direct
proofs, and then we moved on to proofs by contradiction and mathematical induction. The
method of contradiction is an example
SMS1103: TUTORIAL 4
Question 1. Find all positive integers with exactly 2 different prime factors with all prime factors less than
10. Then find all positive integers with exactly 4 different prime factors with all prime factors less than 15.
SMS1103: TUTORIAL 7
System of Linear Congruences
Question 1.  Find solutions of each of the following system of linear equations.
x + 3y
2x + y
2x + y
4x + 2y
2x + 6y
[In which cases, there is unique solution, no soluti
SMS1103: TUTORIAL 2
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?
SMS1103: TUTORIAL 1
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
(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
SMS1103: TUTORIAL 10
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
Chapter 3: Application of
Divisibility Test for power of 2
Divisibility Test for power of 5
Divisibility Test for 3 and 9
Divisibility Test for 11
An Introductory Course in Elementary
These notes serve as course notes for an undergraduate course in number theory. Most if not all universities worldwide offer introductory courses in number
theory for math majors and
SMS1103: TUTORIAL 6
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.