NUMBER SENSE: Factors of Whole Numbers

Divisibility Rules

There are well-known shortcuts for testing if whole
numbers are divisible by other whole numbers.

Practice without calculator
1. Is 324 divisible by 4?
2. Is 324 divisible by 3?
3. Is 324 divisible by 10?

Why does the divisibility rule for 2 work?

Since 10 is divisible by 2, so is 10²
Therefore,

are both whole numbers.

This is true even if 3 in the hundred’s column
and 5 in the ten’s column were replaced by
other numbers.

So, 354 ÷ 2 is a whole number exactly when
the number in the one’s column is divisible by
2. The number in the one’s column must
be 0, 2, 4, 6 or 8.

Why does the divisibility by 3 rule work?
Example: Is 651 divisible by 3?
Yes, because 6 + 5 + 1 = 12 is divisible by
3. Why does this work?

Since any multiple of 99 and any multiple
of 9 are divisible by 3, 651 is divisible by 3
if and only if 6 + 5 + 1 is divisible by 3.

Every composite number > 1 can be written as the
product of primes .

For example:
“Chipaway” technique for 24
24 = 2 • 12
= 2 • 2 • 6
= 2 • 2 • 2 • 3
= 2³ • 3
“Split Asunder” technique for 24
24 = 4 • 6
= 2 • 2 • 2 • 3
= 2³ • 3
“Chipaway” technique for 1200
1200 = 2 • 600
= 2 • 2 • 300
= 2 • 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 for 1200
1200 = 30 • 40
= 5 • 6 • 5 • 8
= 5 • 2 • 3 • 5 • 2 • 2 • 2
= 2⁴ • 3 • 5²

Factor trees can be useful to arrive at prime
factorizations.

Using the “Chipaway” technique for 24

leading to:
24 = 2 • 2 • 2 • 3
= 2³ • 3

Using the “Split Asunder” technique for 24

leading to:
24 = 2 • 2 • 2 • 3
= 2³ • 3

Factors of Whole Numbers - Worksheet

Factor 60 and 500 by drawing factor trees for
both techniques.

“Chipaway” Technique:
60 =
500 =
“Split Asunder” Technique:
60 =
500 =

Fundamental Theorem of Arithmetic :

Every composite number greater than one can
be ex pressed 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.

Except for order, you will obtain the same list
of primes regardless of the method used to
arrive at a prime factorization.

Factors
Factors of 24

So the list of all factors of 24 is
1, 2, 3, 4, 6, 8, 12, 24

Factors of 37
37 = 1 • 37 = 37 • 1
So the list of all the factors is
1, 37

Factors of 64

So the list of factors of 64 is
1, 2, 4, 8, 16, 32, 64
(Note 8 is only listed once)

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

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

Practice
Find the prime factors of 32 and 80.
Find all factors of 32 and 80.

Common Factors
Recalling the following lists of factors
24: 1, 2, 3, 4, 6, 8, 12, 24
64: 1, 2, 4, 8, 16, 32, 64

We see that the common factors of 24 and 64
are:
1, 2, 4 and 8

This leads to the greatest common factor
(GCF):

The GCF of 24 and 64 is 8
or GCF (24, 64) = 8

Practice:
a) Find the GCF(30, 96)
b) Find the GCF(90, 75)

GCF by Prime Factorization

It is also possible to determine GCF(a,b)
from each number’s prime factorization.

Example: Find GCF (360, 270)
360 = 2³ • 3² • 5¹
270 = 2¹ • 3³ • 5¹

Choose the smallest power of each prime
that occurs in either list.

GCF(360, 270) = 2¹ • 3² • 5¹ = 90

Question: How does this work if there are
different primes for each number?
Answer: Use the zero exponent .

GCF and the Zero Exponent
Example: Find GCF(84, 90)
84 = 2² • 3 • 7
90 = 2 • 3² • 5

Rewrite so that all primes appear in both
lists. Use the zero exponent.
84 = 2² • 3¹ • 5⁰ • 7¹
90 = 2¹ • 3² • 5¹ • 7⁰

Now choose each prime raised to the
lowest power.
GCF(84, 90)
= 2¹ • 3¹ • 5⁰ • 7⁰
= 2 • 3 • 1 • 1
= 6

GCF Practice
Find the GCF of 30 and 96 by both methods.
Find the GCF of 90 and 75 by both methods.
Find the GCF of 4500 and 4050 by prime
factorization.