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

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28 Feb 2018

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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

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

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∈ ℤ 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.