S. I. M.
Mr. Plassmann
Section 1.3: Prime Numbers
Objectives:
1.
Be able to identify prime numbers find the prime factorization of any integer.
2.
Use the prime factorization of a two numbers to find their GCD and LCM.
3.
Use Euclid’s Algorithm to find the GCD of two integers.
Vocabulary:
Prime Number
Canonical Decomposition
Prime Factorization
Sieve of Eratosthenes
Relatively Prime
Greatest Common Divisor
Least Common Multiple
Euclid’s Algorithm
P. O. T. D.:
Use the Sieve of Eratosthenes handout to make a list of all primes less than 200.
Lesson:
I. Prime Numbers and Canonical Decomposition
In mathematics, a prime number (or a prime) is a natural number which has exactly two
distinct natural number divisors: one and itself. The prime numbers are unique amongst the
natural numbers because their products form all the other natural numbers. That is the
prime numbers are the building blocks of the integers.
An infinite number of prime numbers exists, as demonstrated by Euclid in about 300 BC.
The property of being a prime is called primality, and the word prime is also used as an
adjective. Since two is the only even prime number, the term odd prime refers to all prime
numbers greater than two. The study of prime numbers is part of number theory, the branch
of mathematics which encompasses the study of natural numbers.
Prime numbers have been the subject of intense research, yet some fundamental
questions concerning them have remained unanswered. The problem of modeling the
distribution of prime numbers is a popular subject for investigation by number theorists:
when looking at individual numbers the primes seem to be randomly distributed, but the
“global” distribution of primes follows well defined laws. It is often useful to find the prime
factorization of an integer, which is also known as the prime or canonical decomposition of a
number.
Example #1: Finding the Canonical Decomposition of an Integer
Find the prime factorization of the following numbers:
A.
B.
504
From the factorization of
504
C.
32,513
457
above, can you determine the number of divisors it has?
That is, how many numbers is it divisible by.
II. Uses of Canonical Decomposition
We can use the prime factorization of integers to find many different properties of two
different integers. Two of the most important are known as the Least Common Multiple
(LCM) and the Greatest Common Divisor (GCD).
Definition of Least Common Multiple:
The Least Common Multiple of a pair of integers
divisible by both
a
and
a
and
b
is the smallest number
c
that is
b.
Definiton of Greatest Common Divisor:
The Greatest Common Divisor of a pair of integers
divides both
a
and
a
and
b
is the largest number
c
that
b.
Example #2: Finding the LCM and GCD of a Pair of Integers
Use the prime factorizations of the integers below to find their LCM and GCD.
A.
84
and
198
B.
110
and
273
Two integers that have a GCD of 1 are said to be relatively prime, that is they have no
prime factors in common.
III.
Euclid’s Algorithm
In number theory, the Euclidean Algorithm (also called Euclid’s Algorithm) is an algorithm
to determine the greatest common divisor of two integers. The algorithm is attributed to
the Greek mathematician Euclid of the third century BC. He is known as “The Father of
Geometry” and is credited with helping to lay the foundation of modern mathematics and
geometry. The major significance of his algorithm is that it does not require factoring the
two integers, and it is also significant in that it is one of the oldest algorithms known. Below
the algorithm is used to find the GCD of
527
and
713 .
713  527 1  186
527  186
2  155
186  155
1  31
155  31
50
Example #3: Using Euclid’s Algorithm to find the GCD of Two Integers
Use Euclid’s Algorithm to find the GCD of
7469
and
2464 .
IV. Homework
Assignment 1.3
PRACTICE PROBLEMS:
1.
Find the prime factorization of the following numbers. Use the prime factorization to
determine the number
A.
2.
B.
623
13,531
Use the prime factorizations of the integers below to find their LCM and GCD.
A.
4.
of divisors of each number:
350
and
805
Use Euclid’s Algorithm to find the GCD of
B.
496
442
and
and
381.
1, 240
S. I. M.
Mr. Plassmann
NAME:____________________
DATE:____________________
BLOCK:___________________
Section 1.3 Homework: Prime Numbers
1. Find the prime factorization of the following numbers. Use the prime factorization to
determine the number
A.
of divisors of each number:
B.
825
22, 064
2. Use the prime factorizations of the integers below to find their LCM and GCD.
A.
312
and
473
3. Use Euclid’s Algorithm to find the GCD of
B.
373
45 , 120 ,
and
3, 473 .
and
75