COUNTING AND NUMBER SYSTEMS
Title
Materials for
Teacher
Materials for
Students
Description
Reflection
Looking Ahead
Multiples and Factors
Multiples and Factors
Overhead projector, handouts, markers
Lesson handouts, colored pens/pencils, graphing
papers, cutout papers/color tiles
We discuss the concept of prime and composite
numbers, factors, and multiples. We also present
techniques in finding the greatest common factor (GCF)
and the least common multiple (LCM) as well as the
basic divisibility rules.
A good understanding of divisibility, greatest common
factor, and least common multiple will be useful in
simplifying and manipulating fractions. Greatest
common factor and least common multiple are usually
taught as a means of combining fractions with unlike
denominators.
1
Introduction
 Number theory is a branch of mathematics that is devoted primarily to the study of
the set of counting numbers.
 Elementary school number theory usually comes right before work with fractions
because simplifying fractions and finding common denominators for adding and
subtracting fractions use number theory ideas.
 In this lesson, we shall study the different aspects of counting numbers that are
useful in simplifying computations, especially those with fractions.
Definition: Factors are the numbers you multiply together to get another number.
Area Model:
To illustrate how to find the factors of a given number, let us look at the following area
model.
Given 12 tiles, arrange the tiles into a rectangular array with no missing corners or gaps
between tiles and without any overlapping tiles.
One way to arrange these 12 tiles is by using 1 row.
We get a 1  12 rectangle.
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2
We can also have it arrange in 2 rows…
Now it is a 2  6 rectangle.
The tiles can also be arranged in three rows…
This is a 3  4 rectangle.
Now note that if we arrange the rectangles with 4 rows…
We get a 4  3 rectangle, which is just a rotation of the 3  4 rectangle, thus giving us the
same rectangle.
Since we already have repeated the arrangement of the rectangle, these would be all the
possible arrangements.
NOTE: If tiles are not available, the students can have a grid paper and they can
color a group of 12 cells/grid boxes to form the rectangles.
Observation:
As you can see from the above arrangements, the rectangle can have the following wholenumber dimensions
1  12
2  6
3  4
Note that 4  3 is the same as the dimensions 3  4 (by commutative property of
multiplication), so we do not write it as a unique representation of the rectangle.
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The whole-numbers 1,2,3,4,6,and 12 that formed the dimensions of the rectangle are the
factors of the number 12, and the factor pairs of 12 are 1 and 12, 2 and 6, and 3 and 4.
Mathematical Algorithm:
The above area model gives us the steps in finding the factors of a whole number. To find
the factors of a whole number follow this procedure:
1. Starting with 1, divide the given whole number by each of the consecutive counting
numbers.
2. If the numbers divide exactly (no remainder), then you have found a pair of factors.
3. List the counting number and the quotient of your division as a pair of factors.
4. Keep dividing until a factor repeats.
5. List all factors separated by commas.
Illustration: Find the factors of 12.
Counting Number
1
2
3
4
Division
12 ÷ 1 = 12
12 ÷ 2 = 6
12 ÷ 3 = 4
12 ÷ 4 = 3
Factor Pair
1 x 12
2x6
3x4
4x3
Solution:
The factors of 12 are 1, 2, 3, 4, 6 and 12. (Note: means that one or more factors have
repeated so we stop dividing.) Again, we see that the factor pairs of 12 are 1 and 12, 2
and 6, and 3 and 4.
Factor pairs can be represented in multiple ways. The multiplication we have on the table
above is one form of writing the pairs.
1  12
2  6
3  4
We can also create factor trees. That is, for factor pairs of 12, we have
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We can also make the T-charts or tables. For instance, for the factor pairs of 12, we can
have
12
1 12
2 6
3 4
Finding the factors of 20.
Counting Number
1
2
3
4
5
Division
20 ÷ 1 = 20
20 ÷ 2 = 10
20 ÷ 3 = 6 R 2
20 ÷ 4 = 5
20 ÷ 5 = 4
Factor Pair
1 x 20
2 x 10
-------4x5
5x4
Solution:
The factors of 20 are 1, 2, 4, 5, 10 and 20. Notice that 3 is not a factor of 20 because 3
does not divide 20 exactly. There is a remainder of 2 when 20 is divided by 3.
Draw the factor trees and T-chart to summarize the factors of 20.
20
1 20
2 10
4 5
The rectangular array can be used to illustrate mathematical relationships between
numbers.
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Consider the 2  6 rectangle.
This rectangular array illustrates the following:
 2  6 = 12
 12 is a multiple of 6 (and of 2)
 12 is the product of 2 and 6.
 2 and 6 are factors of 12.
 2 (and 6) is a divisor of 12.
 12 is divisible by 2 (and by 6).
From the other rectangles, we can see that 1, 3, 4, and 12 are also divisors of 12, since 1
 12 = 12 and 3  4 = 12.
Basic Terminologies:
If m  n = p, then m and n are called factors of p, and p is called a multiple of m (and of
n). We also call p the product of m and n.
If m  n = p and m is not 0, then m is called a divisor of p. We say that p is divisible by
m.

If m is a factor of n, is n a multiple of m?
Answer:

If m = 2  n, what can you say about m in relation to 2?
Answer:
Even and Odd Numbers
An even number is a number that can be divided evenly by 2.
Example: 20 can be divided by 2, so 20 is even. 84 is even because it can be evenly
divided by 2.
Formal definition of an even number:
A number n is even if there is a number k such that n = 2k.
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This is formal way of saying that if n is divided by 2, we always get a quotient k with no
remainder.
Using our area model, this corresponds to saying that the n tiles can be arranged into two
rows and k columns with no missing corners or gaps between tiles and without any
overlapping tiles.
Example: 20 is even because we can arrange 20 tiles into a 2 by 10 rectangle such that
there are no missing corners.
However, 17 is not an even number because we cannot arrange 17 tiles into 2 rows
without missing corners.
Mathematically, 17 cannot be evenly divided by 2. In fact, 17  2 = 8 with a remainder of 1.
If the number cannot be divided evenly by 2, then the number is called an odd number.
Formal definition of an odd number:
A number n is odd if there is a number k such that n = (2  k) + 1.
This is formal way of saying that if n is divided by 2, we always get a quotient k with a
remainder of 1.
As we see in the above example, 17 = (2  8) + 1. Thus, 17 is odd.
How to determine whether a number is even or odd?
To tell whether a number is even or odd, look at the number in the ones place. That single
number will tell you whether the entire number is odd or even.
 An even number has 0, 2, 4, 6, or 8 in the ones place.
 An odd number has 1, 3, 5, 7, or 9 in the ones place.
Example:
Consider the number 3,842,917. It has 7 in the ones place, therefore, 3,842,917 is an odd
number. Likewise, 8,322 is an even number because it ends in 2.
Give an example of a 6-digit number that is even.
Give an example of a 5-digit number that is odd.
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Basic operations with even and odd numbers
Exercise: Perform the following operations on each pair of numbers and determine
whether the result is an even or an odd number.
1. 12 + 26 =
2.
3  407 =
3. 431 + 17 =
4.
18  6 =
5. 93 + 542 =
6.
72  21 =
7. 66 + 484 =
8.
11  19 =
What do you notice about the result of adding two even numbers?
Is the product of two odd numbers an odd number or an even number?
 The sum of two even numbers is even.
Why?
Suppose we have two even numbers, A and B. Using the formal definition of even
numbers, A = 2m and B = 2n.
Then A + B = (2m) + (2n).
By distributive property, we have A + B = 2  (m + n).
Thus, since m + n is a whole number, A + B must be even.

The sum of two odd numbers is even.
Can you show that this is true? (The proof of this is similar to the one above.)

The sum of an even number and an odd number is odd.
Can you prove this one?
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


The product of two even numbers is even.
Why?
Suppose we have two even numbers A and B. Again, by definition of even
numbers, A = 2m and B=2n.
If we take the product AB, we have AB = (2m) (2n) which is equal to
2(2mn), which is even.
The product of two odd numbers is even.
Why?
Suppose we have two odd numbers A and B. By definition of odd numbers, A =
(2m) + 1 and B = (2n) + 1.
If we take the product AB, we have AB = [(2m) + 1] [(2n) + 1] which is equal
to (4mn) + (2m) + (2n) + 2.
By distributive property, we can write the product as 2[(2mn) + m + n + 1], which
is even because the product is a multiple of 2.
The product of an even number and an odd number is odd.
Show that this result holds true.
Summary in a table form:
+
Even Odd
Even Even Odd
Odd Odd Even
Even Odd

Even Even Odd
Odd Odd Odd
Identifying even and odd numbers is an important skill that children will need throughout
their math education. This skill will help prepare them to learn division, prime numbers,
and even square roots.
Divisibility
One strategy to find the factors of a number is by determining if the number is divisible by
another number.
Definition: A whole number is divisible by another number if the quotient is a whole
number and the remainder is zero.
“Divisible by” is the same as “can be evenly divided by”.
Illustration:
14 is divisible by 7 because 14  7 = 2 (EXACT!)
But 16 is not divisible by 7 because 16  7 = 2 with a remainder of 2.
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We shall now discuss a few divisibility rules that we can use to test if one number is
divisible by another, without having to do much calculation.
Why do these divisibility rules work?
We will show why the rule works for some of the above divisibility rules.
In the following discussion, we will discuss the proofs of these divisibility rules based on
the similarity of the test.
Tests for divisibility by 2, 5, and 10
 A number is divisible by 2 if and only if its ones digit is 0,2,4,6, or 8.
 A number is divisible by 5 if and only if its ones digit is 0, or 5.
 A number is divisible by 10 if and only if its ones digit is 0.
Notice that for a number to be divisible by 2, 5, or 10, we only need to look at the
ones/units digit of the number.
Divisibility by 2:
The number is divisible by 2 if the ones/units digit of the number is 0, 2, 4, 6, or 8.
So why does this rule works?
It is clear that if a number is a single digit number, then it will be divisible by 2 if the
number is 0, 2, 4, 6, or 8. [since 0  2 = 0 (no remainder), 2  2 = 1 (no remainder), 4  2 =
2 (no remainder), 6  2 = 3 (no remainder), and 8  2 = 4 (no remainder)]
Now let us consider a three-digit number, say ABC, and the ones/units digit, “C”, is 0, 2, 4,
6, or 8. Then the number can be written as
ABC = 100A + 10B + C
where C = 0, 2, 4, 6, or 8.
We already know that C is divisible by 2. So we need to only show that (100A + 10B) is
divisible by 2 to show that the number “ABC” is divisible by 2.
By distributive property, we can write 100A + 10B as 10(10A + B).
Since 10 is divisible by 2 [102 = 5(no remainder)], any multiple of 10 is divisible by 2.
Thus, 100A + 10B = 10(10A + B) must be divisible by 2 regardless of what the digits
A and B are.
Hence, “ABC” = 100A + 10B + C is divisible by 2.
Note that the reasoning for the divisibility for 5 and for 10 is similar to the divisibility for 2.
Can you show the proof for these two rules?
Divisibility by 5:
The number is divisible by 5 if the ones/units digit of the number is either 0 or 5.
Why does this rule works?
Once again, it is clear that if a number is a single digit number, then it will be divisible by 5
if it is 0 or 5. [since 0  5 = 0 (no remainder) and 5  5 = 1 (no remainder)].
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For similarity of proof, consider a three-digit number, say “ABC”. Can you complete
the proof for this divisibility rule?
Divisibility by 10:
The number is divisible by 10 if the ones/units digit of the number is 0.
Can you show why this rule works? Consider a three-digit number, say “ABC” in
your proof.
Example: Give a 7-digit number that is divisible by 2, 5, and 10.
Next, we look at the divisibility rules for 3 and 9.
Tests for Divisibility for 3 and 9
 A number is divisible by 3 if and only if the sum of its digits is divisible by 3.
 A number is divisible by 9 if and only if the sum of its digits is divisible by 9.
Note that in these two rules, the rule simplifies finding out if a big number is divisible by 3
(or by 9) by reducing the number to a smaller number and determining if the smaller
number is divisible by 3 (or by 9).
Let us see why this rule works.
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Divisibility by 3
A number is divisible by three if the sum of the digits (which is a much smaller number) is
divisible by three.
So why is this rule true?
Consider, for simplicity, a 3-digit number, call it, “ABC”.
Suppose that A+B+C is divisible by 3.
Note that a 3-digit number such as “ABC” is actually 100A + 10B + C.
Let us now rewrite 100A + 10B + C as (99+1) A + (9+1) B + C.
By distributive property, we can write this as 99A + A + 9B + B + C.
By commutative and associative properties of addition, we can rewrite our number as
(99A + 9B) + (A + B + C).
Now observe that 99A + 9B is always divisible by 3 since numbers with all nines are
always divisible by 3. (Observe 33 = 9, 333 = 99, 999 = 3333=, etc.)
Since we assume that A+B+C is divisible by three, then we have a sum of two parts that
are both divisible by three. Hence, the sum must also be divisible by three. So ABC =
(99A + 9B) + (A + B + C) must be divisible by 3.
Let’s show how this works with the number 327.
327 = [(99 x 3) + (9 x 2)]+ [(1 x 3) + (1 x 2) + 7]
Do you understand why we only need to consider the numbers inside the bracket on the
right?
In a similar argument, (that is, by simply showing that 99A + 9B is divisible by 9, we can
prove why the divisibility rule for 9 works.
Can you write the complete proof for the divisibility for 9?
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Divisibility by 9
A number is divisible by 9 if the sum of the digits (which is a much smaller number) is
divisible by 9.
Example: Determine whether 225 is divisible by 3.
Is 225 divisible by 9? Explain your answer.
Now observe that the divisibility rules for 4 and 8 are also very similar…
Tests for divisibility by 4 and by 8
 A number is divisible by 4 if and only if the number represented by its last two digits
is divisible by 4.
 A number is divisible by 8 if and only if the number represented by its last three
digits is divisible by 8.
We will show the proof for the divisibility by 4. But observe that the proof for divisibility by 8
will be similar.
Divisibility by four
A number is divisible by four if the number formed by the last two digits (tens and ones)
are divisible by 4.
This rule simplifies finding out if a big number is divisible by four by checking if the last two
digits is divisible by four.
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So why is this rule true?
Consider, for simplicity, again, a 4-digit number, call it, “ABCD”.
Suppose that CD is divisible by 4.
Note that a 4-digit number such as “ABCD” is actually 1000A + 100B + CD.
Now observe that 1000 and 100 are divisible by 4. So 1000A and 100B are both
divisible by 4. In fact, all powers of 10 greater than or equal to 100 are divisible by 4. (100
= 102 = 254, 1000 = 103 = 2504, etc.,)
Since we assume that CD is divisible by four, then we have a sum that are all divisible by
four. Hence, the sum must also be divisible by four. So ABCD = 1000A + 100B + CD
must be divisible by 4.
Let’s show how this works with the number 6832.
6832 = [(1000 x 6) + (100 x 8)]+ 32
Do you understand why we only need to consider the digits in the tens and ones columns?
Exercise: Show why the test of divisibility for 8 works.
HINT: All powers of 10 greater than or equal to 1000 are divisible by 8 (1000 = 10 3 =
1258, 10000 = 104 = 12508, etc.)
Example: Determine whether 7,168 is divisible by 4.
Remember: In the Gregorian calendar, (the current standard calendar in most of the
world), leap years are years that are divisible by 4.
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Lastly, observe that the divisibility of a number by 6 and by 12 depends on the divisibility of
the number by a pair of factors of 6 and of 12.
Tests for divisibility by 6 and by 12
 A number is divisible by 6 if and only if the number is divisible by both 2 and 3.
 A number is divisible by 12 if and only if the number is divisible by both 3 and 4.
These two rules are specific rules based on the following theorem:
Theorem:
A number is divisible by the product, ab, of two nonzero whole numbers a and b if the
number is divisible by both a and b, and a and b have only the number 1 as a common
factor.
From this theorem, since 6 = 2  3 and 2 and 3 are both prime numbers, thus having only
the number 1 as their common factor, any number that is divisible by 6 must be both
divisible by 2 and by 3.
Similarly, since 12 = 3  4 and 3 and 4 only have the number 1 as their common factor, a
number is divisible by 12 if the number is both divisible by 3 and by 4.
Exercise: Based on the theorem above, devise a way of checking to see whether or not a
number is divisible by
 divisible by 14

divisible by 72
True or False: Determine whether the following statement/s is/are TRUE or FALSE and
explain your answer.
1. If a number is divisible by 3 and by 11, it must be divisible by 33.
2. If a number is divisible by 4 and by 6, then the number is divisible by 24.
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More Exercises on Divisibility:
Example:
1. Determine whether the number 9,042 is divisible by 2,3,4,5,6,8,9,and 10.
9,042 is divisible by 2 since the ones/units digit is 2.
9,042 is divisible by 3 since the sum of the digits
(9+0+4+2=15) is 15, and 15 is divisible by 3.
9,042 is NOT divisible by 4 since 42 is not divisible by 4.
9,042 is NOT divisible by 5 since the last digit is not 0 or 5.
9,042 is divisible by 6 since 9,042 is divisible by both 2 and
3.
9,042 is NOT divisible by 8 since the last 3 digits are 042,
and 42 is not divisible by 8.
9,042 is NOT divisible by 9 since the sum of the digits is 15,
and 15 is not divisible by 9.
9,042 is NOT divisible by 10 since the last digit is not 0 or 5.
Solution:
9,042 is divisible by 2, 3 and 6.
2. Determine whether the number 35,120 is divisible by 2,3,4,5,6,8,9,10, and 12.
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3. Produce a six-digit number such that 2 and 3 are factors of the number, but 4
and 9 are not.
4. What digit could go in the
? Explain your answer.
n = 41,87
, 432
If 3 is a factor of n, give all the single-digit possibilities that could go in
the
.
What could go in the
n = 41,87
, 432
Multiples and Factors
if 9 is a factor of n? Explain your answer.
17
Below is a summary of all the divisibility rules we have discussed. The divisibility rule for 7
and 11 are included in the chart below.
A number is
divisible by:
2
3
4
If:
The ones digit is 0,2,4,6, or 8
The sum of the digits is
divisible by 3
The number formed by the last
2 digits (tens and ones digits) is
divisible by 4.
5
The ones digit is 0 or 5
6
The number is divisible by both
2 and 3
7
8
If you double the last digit and
subtract it from the rest of the
number and the answer is:
• 0, or
• divisible by 7
(Note: you can apply this rule to
that answer again if you want)
The number formed by the last
3 digits (hundreds, tens and
ones digits) are divisible by 8.
The sum of the digits is
divisible by 9
9
10
(Note: you can apply this rule to
that answer again if you want)
The ones digit is 0.
Example:
128 is divisible by 2 (ends in 8)
129 is not divisible by 2 (ends in 9)
381 (3+8+1=12, and 12÷3 = 4) Yes
217 (2+1+7=10, and 10÷3 = 3 1/3) No
1312 is (12÷4=3)
7019 is not
175 is
809 is not
114 (it is even, and 1+1+4=6 and 6÷3 =
2) Yes
308 (it is even, but 3+0+8=11 and 11÷3
= 3 2/3) No
672 (Double 2 is 4, 67-4=63, and
63÷7=9) Yes
905 (Double 5 is 10, 90-10=80, and
80÷7=11 3/7) No
109816 (816÷8=102) Yes
216302 (302÷8=37 3/4) No
1629 (1+6+2+9=18, and again, 1+8=9)
Yes
2013 (2+0+1+3=6) No
220 is
221 is not
11
If you sum every second digit
and then subtract all other
digits and the answer is:
• 0, or
• divisible by 11
1364 ((3+4) - (1+6) = 0) Yes
3729 ((7+9) - (3+2) = 11) Yes
25176 ((5+7) - (2+1+6) = 3) No
12
The number is divisible by both
3 and 4
648 (6+4+8=18 and 18÷3=6, also
48÷4=12) Yes
916 (9+1+6=16, 16÷3= 5 1/3) No
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Divisibility tests are useful in investigating whether a given number is a prime.
Prime and Composite Numbers
Area Model
Now, suppose we have 11 tiles. Try to arrange the 11 tiles into a rectangle. Again, make
sure that the tiles are arranged into a rectangular array with no gaps between tiles, no
missing corners, and no overlapping tiles.
If we form 3 rows of tiles…
There is a missing corner in the bottom! So this dimension would not work.
If we use 2 rows of tiles…
There is again a missing corner! So this would not do it either.
But if we only make a rectangle with just one row…
We get a perfect fit! No missing corner! Thus, 1  11 rectangle works!
And this is the only way to arrange the tiles that will satisfy our conditions.
Observation:
 Since the eleven tiles can only be arranged into a rectangle in only one way, that is,
in a 1  11 rectangle, 11 only has 1 and itself (that is, 11) as its factors.
Definition: Numbers with only two factors are known as prime numbers, and the only two
factors are one (1) and the number itself.
If a number has more than two factors, we say that the number is composite.
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Illustrations:



11, as seen in the above area model, is prime because its only factors are 1 and itself
(11).
12, on the other hand, is a composite number because 12 have other factors other
than 1 and 12, such as, 2, 3, and 4, as we have seen in the earlier discussion.
1 is neither prime nor composite because it has only one factor. According to the
definition, a number must have at least two factors before it can be identified as prime
or composite.
Question: Is 13 a prime number?
Solution: 13 is a prime number. Using the method of finding factors above, we can show
that 13 can only be divided evenly by 1 and 13.
Thus, an alternative definition to a prime number is a number that can be divided evenly
only by 1 and itself.
Can you list the first ten prime numbers?
Now note that 4 is not a prime number. Let’s investigate why…
In terms of area models, we can create the following rectangles from 4 tiles.
This is a 1  4 rectangle (equivalent to a 4  1 rectangle) and
this is a 2  2 rectangle.
We cannot arrange the 4 tiles into three rows without missing corners.
Thus, 4 has three factors, namely 1,2, and 4.Hence, 4 is not a prime number. 4 is a
composite number.
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Using our mathematical method…
Counting Number
1
2
3
4
Division
4÷1=4
4÷2=2
4÷3=1R1
4÷4=1
Factor Pair
1x4
2x2
-------4x1
We also obtain the same result. That is, 4 is a composite number, not a prime number.
Prime numbers have exactly two factors. We see in the above example that the number 4
has three factors. Find other numbers that have exactly three factors. What do these
numbers have in common?
HINT: Look for numbers with three factors, not three prime factors. The number itself and
1 are always factors, so there must be exactly one other factor. When we factor a number,
we typically get two distinct factors. How could we get only one new factor?
Why are prime numbers important?
Prime numbers are viewed as the basic building blocks of all numbers.
The composite numbers are made up of prime numbers multiplied together, which is why
they are called “composite” numbers, because composite means “something made by
combining things”.
Equivalently, this means we can “break apart” composite numbers into prime number
factors. This method is called prime factorization.
Definition: The prime factorization of a number is a product of primes that equals the
number.
Mathematical Model:
To express a number using prime factorization, divide the number by its smallest prime
factor and repeat until the last quotient is a prime. The product of the prime factors used
as divisor and the prime quotient is the prime factorization of the number.
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Example: Find the prime factors of 18.
Solution:
First divide 18 by 2 (since 2 is the smallest prime factor of 18).
18  2 = 9.
Since 9 is not a prime number, we divide it again by its smallest prime factor. In the case
of 9, 3 is its smallest prime factor (not 2).
9  3 = 3.
Since 3 is a prime number, we stop at this stage.
Hence, 18 = 2  3  3.
Area Model:
We can visualize the above process using tiles.
We begin with 18 tiles. We arrange them into the smallest number of rows (>1) for which
we get a rectangle with no missing corner/s.
Here we get a 2  9 rectangle. Since the number of columns is not yet a prime number, we
collect 9 tiles and arrange them again into the smallest number of rows (>1).
This gives us a 3  3 rectangle. Since the number of columns is also a prime number (that
is, 3), we stop here and combine our result. Since 18 = 2  9 and 9 = 3  3. Then we can
write 18 as 2  3  3.
This area model above can be presented also using a factor tree. That is,
Multiples and Factors
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NOTE: There is only one (unique!) set of prime factors for any number. (Fundamental
Theorem of Arithmetic)
A different order, such as 6 = 2  3 and 6 = 3  2 does not count as a different way
(because of the commutative property of multiplication).
For consistency and easy comparison of prime factorization, practice writing the prime
factors in increasing order.
Although each whole number has a prime factorization, when a number is very large,
finding its prime factorization may be hard. We can use the divisibility tests to find the
prime factorization of large numbers.
Example: Consider n=12,320. What is its unique prime factorization?
We know that 2 divides the number because it ends in 0. We also know that 5 divided
the number because it ends in 0. So we also know that 2  5 = 10 is a factor of
12,320. Thus, we can write 12,320 = 2  5  1232.
We also know that 4 divides 1232 because 32 (the number formed by the last two
digits of 1232) is divisible by 4. Dividing 1232 by 4, we get 308. So now we have
12,320 = 2  5  4  308 = 2  2  2  5  308.
But 4 is also a factor of 308 (since 08 or 8 is divisible by 4). And since 308 = 4  77,
we now have
12,320 = 2  2  2  5  4  77 = 2  2  2  2  2  5  77.
Lastly, 77 is equal to 7  11, which are both prime numbers. Thus, we finally have the
prime factorization of 12, 320.
12,320 = 2  2  2  2  2  5  7  11
We can write the above prime factorization more compactly using exponents as
12,320 = 25  5  7  11.
Practice Problems:
1. Find the prime factorization of 1224 using the above method.
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23
2. Find the prime factorization of 4620 using the above method.
3. Is the number 91 prime or composite? Use divisibility when possible to find the
answer.
Greatest Common Factor and Least Common Multiple
An understanding of factors and multiples is important in order to understand the meaning
of Greatest Common Factor (GCF) and Least Common Multiple (LCM).
Recall that if m  n = p, then m and n are called factors of p, and p is called a multiple of
m (and of n).
DEFINITION: The greatest common factor (GCF) of two numbers is the greatest number
that is a factor of both numbers. The GCF of two numbers a and b is written GCF(a,b).
On the other hand, the least common multiple (LCM) of two numbers is the smallest
number that is a multiple of both numbers. The LCM of a and b is written as LCM(a,b).
REMARKS:
 Since 1 is a factor of each number, all numbers have at least one common factor.
 The GCF is also called the greatest common divisor (GCD) because when a
number is a factor, it is also a divisor. The GCF is often used when simplifying
fractions.
 Since the product of the two numbers is a multiple of both numbers, there is at least
one common multiple for any pair of numbers.
 The LCM is used to find the least common denominator (LCD) when adding or
subtracting fractions.
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AREA MODEL:
Suppose we have a 24 × 36 rectangle and we are interested in finding the largest square
that could tile the entire rectangle without gaps or overlaps.
Start with the 24 × 36 rectangle.
The largest square tile that fits inside this rectangle and is flush against one side is 24 by
24. Only one tile of this size will fit:
The largest square tile that fits inside the remaining rectangle and is flush against one side
is 12 by 12. Two tiles of this size will fit. The original rectangle is now completely filled:
Note that the 24-by-24 square could also be filled with the 12-by-12 tiles, so 12 by 12 is
the largest tile that could fill the original 24-by-36 rectangle.
Multiples and Factors
25
DISCUSSION: The dimensions of a square that would tile that entire rectangle could be a
common factor of the dimensions of the rectangle.
Thus, the GCF would be the dimensions of the largest square that could tile the entire
rectangle without gaps or overlap.
In the above model, since 12 is the dimension of the largest square that tiles the entire 24
by 36 rectangle, we say that 12 is the GCF of 24 and 36.
Now suppose that we have a 24 by 36 rectangular tile that could tile a square. Let us find
the smallest square that could be tiled by such rectangles.
Let us start with the 24-by-36 rectangle. Our goal is to make a square tiled with rectangles
of these dimensions:
Since the width (24) is less than the height (36), add a column of tiles to the right of the
rectangle (in this case, one tile). This makes a 48-by-36 rectangle:
The width (48) is now greater than the height (36), so add a row of tiles under the existing
rectangle (in this case, two tiles). This makes a 48-by-72 rectangle:
Multiples and Factors
26
The width (48) is now less than the height (72), so add another column (two tiles) to the
right of the existing rectangles. The dimensions are now 72 by 72 -- and you've made a
square!
The 72-by-72 square is the smallest square that can be tiled with a 24-by-36 rectangle.
DISCUSSION: Any common multiple of 24 and 36 could be the dimensions of a square
that could be tiled by this rectangle. For example, since 24 × 36 = 864, a square that is
864 by 864 could be tiled by the 24-by-36 rectangle.
Thus, the LCM of 24 and 36 would be the dimensions of the smallest square that could be
tiled by the 24-by-36 rectangle. Since a 72-by-72 square is the smallest square that can be
tiled using 24-by-36 rectangular tiles, the LCM of 24 and 36 is 72.
Prime Factorization Method:
Finding the prime factorization of the two given numbers can be a good start to find either
the greatest common factor (GCF) or the Least Common Multiple (LCM).
First let us factor 24 and 36 into their prime factors.
From these, we see that 24 = 2 × 2 × 2 × 3 and 36 = 2 × 2 × 3 × 3.
I will line up these factors out, all nice and neat, with the factors lined up according to
occurrence:
24:
36:
2
2
2
2
2
3
3
3
Writing this orderly listing, with each factor having its own column will do most of the
work.
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Finding GCF using prime factorization:
Let us find the GCF of 24 and 36 using prime factorization.
The GCF of two numbers is the biggest number that will divide both 24 and 36. In other
words, it is the number that contains all the factors that is common to both numbers.
Looking at the nice listing, we can see that 24 and 36 have a pair of 2 in common and a 3.
Thus,
24:
36:
GCF:
2
2
2
2
2
2
2
3
3
3
3
Thus, the GCF of 24 and 36 (written as GCF(24,36)) is the product of these three
common prime factors. That is, GCF(24,36) = 2×2×3 = 12. (agrees with the area model.)
Finding LCM using prime factorization:
Let us find the LCM of 24 and 36 using prime factorization.
The least common multiple (LCM) is the smallest number that contains both 24 and 36.
Then it would be the smallest number that contains one of every factor in these two
numbers.
Looking back at the list, I see that 24 has 3 copies of 2 and 36 has 2 copies of 2. Since the
LCM must contain all factors of each number, the LCM must contain all three copies of 2.
Similarly, since there are 2 copies of 3 in 36 and only 1 copy of 3 in 24, the LCM must
contain 2 copies of 3.
24:
2 2 2 3
36:
2 2
3 3
LCM: 2 2 2 3 3
Thus, the LCM of 24 and 36 is 2 × 2 × 2× 3 × 3 = 72. (agrees with the result of the area
model.
In summary, by listing the prime factors neatly in a table, you can find the GCF by carrying
down only those factors that the numbers share and multiplying them. For the LCM, you
simply carry down all the factors listed in each column, regardless of how many or how
few values contained that factor in their listing.
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Venn Diagram Model:
An alternative way to explore the common factors and multiples of the two numbers is to
use a Venn diagram.
If we first separately draw the circles representing each number and write down the prime
factors for each number inside the circles, we get
The circle on the left contains all the prime factors of 24, and the circle on the right
contains all the prime factors of 36.
Observe that the factors 2, 2, and 3 are contained in both circles.
Let us redraw the picture so that the two circles overlap where the common factors are
located.
The greatest common factor (GCF) of 24 and 36 is the product of all numbers inside the
overlapping section of the two circles. That is, 2  2  3, or 12.
On the other hand, the least common multiple (LCM) of 24 and 36 is the product of all
numbers in the circles. That is, 2 (from the circle for 24)  (2  2  3) (the numbers in the
overlap portion)  3 (from the circle for 36), or 2  2  2  3  3 = 72. Thus, the least
common multiple (LCM) of 24 and 36 is 72.
PRACTICE: Find the GCF and LCM of 42 and 90 using the area model, the prime
factorization model and the Venn diagram model.
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MORE PRACTICE: Use the area model, Venn diagram model, and prime factorization
model to find the GCF and LCM of the following:
a. 30 and 42
b. 18 and 30
The above prime factorization method can be extended to three or more numbers.
EXAMPLE: Find the GCF and LCM of 27, 90, and 84.
Listing the prime factors of 27, 90, and 84 in a table, we have
27:
3 3 3
90
2
3 3
5
84:
2 2 3
7
GCF:
3
LCM: 2 2 3 3 3 5 7
Since 3 is the only factor that is shared by all three numbers, the GCF of 27, 90, and 84 is
3, while its LCM is 2× 2 × 3 × 3 × 3 × 5 × 7 = 3,780.
PRACTICE: Find the GCD and LCM of 27, 36, 45, and 60.
NUMBER LINE MODEL:
For pairs of small numbers, we can also determine their least common multiple using the
number line.
Suppose we want to find the least common multiple of 3 and 5. Clearly, 3 and 5 are both
prime numbers, so their least common multiple is their product, 3 × 5 = 15.
Using the number line, we can find the LCM by using the fact that the multiples of a
number, are the numbers that you get when you skip count. Thus, to find the LCM, we skip
counting to write the multiples of the first number and skip count to write the multiples of
the second number up to their product. Then search for the first pair that the two skips
meet, this will be the least common multiple of the numbers.
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30
In our example, we have
and we see that the first time the two skips met was at the product of 3 and 5, thus, their
least common multiple is 15.
EXAMPLE: Using the number line method, find the LCM of 6 and 9.
Lastly, there is an interesting relationship between the GCF and the LCM of two numbers.
THEOREM: If a and b are two whole numbers. Then
a  b = GCF(a,b)  LCM(a,b)
This theorem can be used to find the LCM if the GCF of two numbers is known.
From this theorem, we can find the LCM(a,b) as the quotient of the product of a and b and
their GCF. That is,
ab
LCM (a, b) 
.
GCF (a, b)
EXAMPLE: Suppose we want to find the LCM of 36 and 56.
SOLUTION:
The GCF of 36 and 56 is 4. Therefore, the LCM of 36 and 56 is
LCM (36,56) 
Multiples and Factors
36  56
36  56

 9  56  504.
GCF (36,56)
4
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