MA121 Study Guide - Final Guide: Chinese Remainder Theorem, Right Triangle, Hypotenuse

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Wilfrid Laurier University
MA 121
Introduction to Mathematical Proofs
Fall 2017
Final Exam Review Notes
Prof: Robert John Rundle
Exam Guide
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Table of Contents
CHAPTER 5: NUMBER THEORY
5.4 Least Common Multiple and Congruence's of integers
Arithmetic of Congruence's of integers
CHAPTER 4: COMPLEX NUMBERS
6.2 Complex Numbers
6.4 Argand Diagrams
6.5 Trigonometric Functions and Radians
6.6 The Polar and exponential Forms
6.7 Multiplication, Division in polar and Exponentiation
6.8 Roots of a Complex Number
ADDITIONAL MATERIALS
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CHAPTER 5: NUMBER THEORY
5.4 LEAST COMMON MULTIPLE AND CONGRUENCE'S OF INTEGERS
Recall
The integers a and b are said to be relatively prime iff
( )
1,gcd =ba
. In other words a and b are
relatively prime iff they have no proper common factors.
a, b ∈ℤ are relatively prime iff there exists k, l such that +=1.
There is no largest prime number (i.e. there are infinitely many distinct primes).
Definition: Least Common Multiple (LCM)
If a, b ∈ℤ are both not 0 then the least common multiple of a and b is the smallest positive
integer which is a multiple of both a and b, and is denoted by [, ]
When you’re looking for the least common multiple (LCM), you’re trying to find the smallest
number that is a common multiple for two or more numbers.
How do you do this? There are two ways.
Example
Method 1
Here is the longer, but more straightforward way. You can make lists.
Question one
Find the least common multiple for 4, 10 and 12:
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60...
Multiples of 10: 10, 20, 30, 40, 50, 60…
Multiples of 12: 12, 24, 36, 48, 60…
LCM = 60
Question Two
What is the LCM of 8 and 12
Multiples of 8: 8, 16, 24, …
Multiples of 12: 12, 24, …
Thus the LCM of 8 and 12 is 24.
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