NUMBER SENSE: Factors of Whole Numbers

The Winning EQUATION
A HIGH QUALITY MATHEMATICS PROFESSIONAL DEVELOPMENT
PROGRAM FOR TEACHERS IN GRADES 4 THROUGH ALGEBRA II

STRAND: NUMBER SENSE : Factors of Whole Numbers

MODULE TITLE: PRIMARY CONTENT MODULE IV

MODULE INTENTION: The intention of this module is to inform and instruct participants in
the underlying mathematical content in the area of factoring whole numbers.

THIS ENTIRE MODULE MUST BE COVERED IN-DEPTH.
The presentation of these Primary Content Modules is a departure from past professional
development models. The content here, is presented for individual teacher’s depth of
content in mathematics. Presentation to students would, in most cases, not address the
general case or proof, but focus on presentation with numerical examples.

TIME :2 hours
PARTICIPANT OUTCOMES:
•Demonstrate understanding of factors of whole numbers.
•Demonstrate understanding of the principles of prime and composite numbers.
•Demonstrate how to determine the greatest common factor and the least common
multiple when relating two whole numbers.

PRIMARY CONTENT MODULE IV
NUMBER SENSE: Factors of Whole Numbers

Facilitator’s Notes

Ask participants to take the pre-test. After reviewing the results of
the pre-test proceed with the fol lowing lesson on factors of whole
numbers.

Divisability: While expressions of the form a = b • q + r enable us to
divide any whole number a by any counting number b, assuming a >
b, special importance is attached to the case when r = 0. A number b
> 0 is called a divisor or factor of a if there exists a whole number q
so that a = b • q.

Since 0 = b • 0, 0 is divisible by any b > 0.

Whenever a > 0 and a = b • q, there is a rectangular array of b • q dots
that corresponds to the factorization of a by b.

Zero and One : Note that,
a = 1 • a or a ÷ 1 = a
and
0 = a • 0 or 0 ÷ a = 0

leads to the conclusion that 0 is divisible by any number a ≠ 0 and
any number a is divisible by 1.

Caution: Division by zero is not allowed.

Suppose a ÷ 0 = q Then a = 0 • q.

This is impossible if a ≠ 0. If a = 0, any value of q works, but for
division there can only be one answer.

An example is a = b • q, assuming a > b,

24 = 4 • 6
a factors as b • q
24 factors as 4 • 6

Have participants think about how to represent 24 as factors another
way, and how to represent factors of 7.

Factoring is an important skill for later applications (e.g., it arises in
the addition of fractions) and a concept of interest in its own right.
An engaging way to introduce students to this topic is to relate it to
the study of prime numbers.

A whole number is said to be prime if it has exactly two factors: one
and itself. This definition keeps 1 from being a prime. A number > 1
that is not prime is called a composite. The first of these definitions
leads to the following list of primes:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,…

Primes have been studied for thousands of years, and many of their
important properties were established by the ancient Greeks. Among
these is the fact that the above list goes on indefinitely; i.e., there is
no largest prime.

Show participants short cuts for testing if whole numbers are divisible
by other whole numbers.

Use appendix section on justification for divisibility rules at this
point, if desired.

Every whole number > 1 can be written as the product of prime
factors.

For example:
“Chipaway” technique
24 = 2 • 12
= 2 • 2 • 6
= 2 • 2 • 2 • 3
= 2³ • 3

“Split Asunder” Technique
24 = 4 • 6
= 2 • 2 • 2 • 3
= 2³ •

“Chipaway” technique
1200 = 2 • 600
= 2 • 2 • 300
= 2 • 2 • 150 = 2 • 2 • 2 • 2 • 75
= 2 • 2 • 2 • 2 • 3 • 25
= 2 • 2 • 2 • 2 • 3 • 5 • 5
= 2⁴ • 3 • 5²

“Split Asunder” Technique
1200 = 30 • 40
= 5 • 6 • 5 • 8
= 5 • 2 • 3 • 5 • 2 • 2 • 2
= 2⁴ • 3 • 5²

These techniques may be accomplished with the process of factor
trees. Transparency T-8 demonstrates this process.

Have participants use both techniques on 60 and 500 and draw the
corresponding factor tree.

The Fundamental Theorem of Arithmetic asserts that except for
order, you will obtain the same list of primes regardless of the method
used to arrive at a prime factorization.

It states that every composite number greater than one can be
expressed as a product of prime numbers. Except for the order in
which the prime numbers are written, this can only be done in one
way.

Factors and GCFs: Aside from being able to factor whole numbers
into products of primes, it is also important to be able to develop lists
containing all factors (prime and composite) of a particular number.

For example,

The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.
The factors of 37 are: 1, 37.
The factors of 64 are: 1, 2, 4, 8, 16, 32, 64.

Except in the case of perfect squares , factors appear in pairs whose
product is the number being factored.

Have participants find the factors of 32 and of 80 using H-11. Have
participants discuss the difference between “finding all factors” and
“finding the prime factorization.” Example: A list of all factors
of 80 is 1, 2, 4, 5, 8, 10, 16, 20, 40, 80 because 80 is divisible by all of
these. The prime factorization of 80 is 2 • 2 • 2 • 2 • 5 = 2⁴ • 5
because 2 and 5 are both primes.

Given two numbers such as 24 and 64, we can form a list of common
factors. Referring to the above lists of factors of 24 and 64, we
conclude that

The common factors of 24 and 64 are 1, 2, 4, and 8.

The last list leads to the important concept of greatest common
factor (GCF):

The greatest common factor of 24 and 64 is 8.
This is sometimes written as GCF(24, 64) = 8.

A) List of factors
30: 1, 2, 3, 5, 6, 10, 15, 30
96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96

Common factors of 30 and 96 are 1, 2, 3, 6
GCF(30, 96) is 6.

B) List of factors
90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
75: 1, 3, 5, 15, 25, 75

Common factors of 90 and 75 are 1, 3, 5, 15
GCF(90, 75) is 15.

It is possible to determine GCF(a,b) from the prime factorizations of
a and b. Follow the examples on the slides.

A) 30 = 2 • 3 • 5 = 2 • 3 • 5
96 = 2 • 2 • 2 • 2 • 2 • 3 = 2⁵ • 3

GCF(30, 96) = 2 • 3 = 6

B) 90 = 2 • 3 • 3 • 5= 2 • 3² • 5
75 = 3 • 5 • 5 = 3 • 5²

GCF(90, 75) = 3 • 5 = 15

C) 4500 = 2² • 3² • 5³
4050 = 2¹ • 3⁴ • 5²

GCF(4500, 4050) = 2¹ • 3² • 5² = 450

Closely related to GCF is the concept of least common multiple
(LCM). A somewhat awkward way of finding LCM(30, 96) is to
write lists of multiples of 30 and of 96. Surely 30 • 96 is on both of
these lists. We are, however, looking for the smallest number
common to both lists.

Here we find:

Multiples of 30:
30, 60, 90, 120, 150,…, 450, 480, 510,…, 2880, 2910,…
Multiples of 96:
96, 192, 288, 384, 480, 776,…, 2880, 2976,…

so that LCM(30, 96) = 480.

However, LCM(30, 96) can also be found as the smallest product of
primes that contains the prime factorizations of both 30 and 96.
Recalling that 30 = 2 • 3 • 5 and that 96 = 2 • 2 • 2 • 2 • 2 • 3 = 2⁵ • 3,
we find that

LCM(30, 96) = 2⁵ • 3 • 5 = 480

Have participants find LCM(24, 64) and LCM(32, 48) using prime
factorization.

The characterization of GCF and LCM in terms of prime factorization
leads to the following important fact:

GCF(a, b) • LCM(a, b) = a • b.