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Math in Our World
Section 5.1
The Natural Numbers
Learning Objectives



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Find the factors of a natural number.
Identify prime and composite numbers.
Find the prime factorization of a number.
Find the greatest common factor of two or more
numbers.
 Find the least common multiple of two or more
numbers.
Natural Numbers
The set of natural numbers, also known as
counting numbers, consists of the numbers
1, 2, 3, 4, . . . .
Every natural number can be written as the product of two or
more natural numbers. For example,
12 = 3 x 4 16 = 4 x 4 19 = 1 x 19 30 = 2 x 3 x 5
The natural numbers that are multiplied to get a product are
called the factors of that product. We can also write 12 as
1 x 12 or 2 x 6. Now we have a list of all factors of 12: 1, 2,
3, 4, 6, and 12.
EXAMPLE 1
Finding Factors
Find all factors of 24.
SOLUTION
Think of all the ways you can write 24 as a product of two
numbers, starting with 1 as one of the factors, and
working upward.
1 x 24
2 x 12
3x8
4x6
These are all of the possibilities, so the factors of 24 are
1, 2, 3, 4, 6, 8, 12, and 24.
Divisibility
A natural number a is divisible by another natural
number b if dividing a by b results in a remainder of
zero. In this case, we will write b  a. This is also
read as “b divides a.”
Six is a factor of 24 because 24 ÷ 6 = 4 with no remainder,
but 5 is not a factor of 24 because 24 ÷ 5 has remainder 4.
For this reason, we will sometimes call the factors of a
number its divisors.
We know that 6 is a factor of 24, so this means that 24 is
divisible by 6, and we would write 6  24. (This should not
be confused with 6/24, which is the number 6 divided by 24.)
Prime and Composite Numbers
A natural number is called prime if it has
exactly two factors, 1 and itself.
Most numbers have more than two factors, but not all: for
example, 7 can only be written as 1 x 7. So 7 is a prime
number.
A natural number is called composite if it
has three or more factors.
EXAMPLE 2
Deciding if a Number is Prime
Decide whether each number is prime or
composite.
(a) 25
(b) 17
(c) 12
(d) 31
EXAMPLE 2
Deciding if a Number is Prime
SOLUTION
If we can find even one number other than 1 and the
number itself, then it is composite.
(a) Five is a factor of 25, since 5 x 5 = 25, so 25 is
composite.
(b) The only factors of 17 are 1 and 17, so it is prime.
(c) Two is a factor of 12 (as are several other numbers), so
12 is composite.
(d) There are no factors of 31 other than 1 and 31, so it is
prime.
The Fundamental Theorem of
Arithmetic
Every composite number can be written as the
product of prime numbers, and there’s only one way
to do so. (The order of the factors is unimportant.)
We know that there’s more than one way to write 12 as a product:
for example, we could write it as 12 = 2 x 6, or 12 = 3 x 4. But
notice that some of those factors can also be written as products.
Now let’s rewrite each multiplication statement:
12 = 2 x 6 = 2 x 2 x 3
12 = 3 x 4 = 3 x 2 x 2
We ended up with the same result (aside from the order, which
doesn’t matter), and all of the factors are prime numbers.
EXAMPLE 3
Finding Prime Factorization
Using the Tree Method
Find the prime factorization of 100 using the tree
method.
SOLUTION
Start with any factorization of 100, say 2 x 50, then factor
50 as 5 x 10. Finally factor 10 as 2 x 5. This is shown using
a tree.
Rearrange the factors in order: 2 x 2 x 5 x 5 or 22 x 52.
EXAMPLE 4
Finding Prime Factorization
Using the Division Method
Find the prime factorization of 100 using the
division method.
SOLUTION
First, divide 100 by 2 and then
divide the answer by 2.
Continue dividing the answer
until you cannot find an
answer that is divisible by 2,
then try to divide by 3, then 5,
etc., as shown.
This shows that 100 = 2 x 2 x 5 x 5 = 22 x 52.
25 is not
divisible by
3, so we
move on to
5.
Greatest Common Factors
The greatest common factor of two or more
numbers is the largest number that is a factor of
all of the original numbers. We will use the
abbreviation GCF to represent the greatest
common factor.
Let’s look at the factors of 18 and 24:
18: 1, 2, 3, 6, 9, 18
24: 1, 2, 3, 4, 6, 8, 12, 24
These two numbers have four factors in common: 1, 2, 3,
and 6. Obviously, the largest of these is 6, so we will call
it the GCF of 18 and 24.
Greatest Common Factors
Procedures for finding the GCF of two or
more numbers:
Step 1 Write the prime factorization of each
number.
Step 2 Make a list of each prime factor that
appears in all prime factorizations. For prime
factors with exponents, choose the smallest
power that appears in each.
Step 3 The GCF is the product of the numbers
you listed in Step 2.
EXAMPLE 5
Finding the Greatest Common
Factor of Two Numbers
Find the GCF of 72 and 180.
SOLUTION
Step 1 Write the prime factorizations of 72 and 180:
72 = 2 x 2 x 2 x 3 x 3 = 23 x 32
180 = 2 x 2 x 3 x 3 x 5 = 22 x 32 x 5
Step 2 List the common factors: 2 (with exponent 2) and 3
(with exponent 2).
Step 3 The GCF is 22 x 32 = 36.
EXAMPLE 6
Finding the Greatest Common
Factor of Three Numbers
Find the GCF of 40, 60, and 100.
SOLUTION
Step 1 Write the prime factorizations of 40, 60, and 100:
40 = 2 x 2 x 2 x 5 = 23 x 5
60 = 2 x 2 x 3 x 5 = 22 x 3 x 5
100 = 2 x 2 x 5 x 5 = 22 x 52
Step 2 List the common factors: 2 (with exponent 2) and 5
(with exponent 1).
Step 3 The GCF is 22 x 5 = 20.
EXAMPLE 7
Applying the GCF to
Packaging Goods for Sale
An enterprising college student gets a great deal on
slightly past-their-prime packets of instant coffee from
a chain of coffee stores. He acquires 200 packets of
decaf and 280 packets of regular coffee. The plan is to
package them for resale at a college fair so that
there’s only one type of coffee in each box, and every
box has the same number of packets. How can he do
this so that each box contains the largest number of
packets possible? (Of course, he wants to package all
of the coffee he has.)
EXAMPLE 7
Applying the GCF to
Packaging Goods for Sale
SOLUTION
The number of packets per box has to be a factor of both
200 and 280 so that there won’t be any left over. If the
boxes are to contain the largest number of packets
possible, our entrepreneur will need to find the greatest
common factor.
200 = 2 x 2 x 2 x 5 x 5 = 23 x 52
280 = 2 x 2 x 2 x 5 x 7 = 23 x 5 x 7
The factors common to both numbers are 2 (with exponent
3) and 5 (with exponent 1). So the GCF is 23 x 5 = 40. That
tells us that he should put 40 packets in each box.
He’ll have 5 boxes of decaf and 7 boxes of regular to sell.
Least Common Multiples
The least common multiple (LCM) of two or
more numbers is the smallest number that is a
multiple of each. If you like, you can think of it as
the smallest number that is divisible by all of the
numbers.
If you multiply any natural number by 1, then 2, then 3, and so
on, you will generate a list of numbers. We call this list the
multiples of a number. Take a look at the multiples of 4 and 6:
Multiples of 4: 4, 8, 12, 16, 20, 24, . . .
Multiples of 6: 6, 12, 18, 24, 30, 36, . . .
The smallest number on both lists is 12, so we will call it the
LCM of 4 and 6.
Least Common Multiples
Procedures for finding the LCM of two or
more numbers:
Step 1 Write the prime factorization of each
number.
Step 2 Make a list of every prime factor that
appears in any of the prime factorizations. For
prime factors with exponents, choose the largest
that appears in any factorization.
Step 3 The LCM is the product of the numbers
you listed in Step 2.
EXAMPLE 8
Finding the Least Common
Multiple of Three Numbers
Find the LCM of 24, 30, and 42.
SOLUTION
Step 1 Write the prime factorizations of 24, 30 and 42:
24 = 2 x 2 x 2 x 3 = 23 x 3
30 = 2 x 3 x 5
42 = 2 x 3 x 7
Step 2 List all the factors that appear: 2 (with largest
exponent 3), 3, 5 and 7.
Step 3 The LCM is 23 x 3 x 5 x 7 = 840.
EXAMPLE 9
Applying the LCM to
Grocery Shopping
Have you ever noticed that many hot dogs come in
packages of 10, but most hot dog buns come in
packages of eight? (Who thought that was a good
idea?)
What’s the smallest number of packages you can
buy of each so that you end up with the same
number of hot dogs and buns?
EXAMPLE 9
Applying the GCF to
Grocery Shopping
SOLUTION
The total number of hot dogs we buy will be a multiple of
10, while the number of buns will be a multiple of 8. What
we need to find is the least common multiple of 8 and 10.
So we write the prime factorization of each:
8 = 2 x 2 x 2 = 23
10 = 2 x 5
The factors that appear in either list are 2 (with largest
exponent 3) and 5, so the LCM is 23 x 5 = 40. That means
we would need four packages of hot dogs and five
packages of buns. Hope you’re hungry!
An Alternative Method for
Finding the GCF
The Divide by Primes Method for Finding the GCF
of two or more numbers:
Step 1 List the numbers horizontally.
Step 2 Find a prime number that divides all of the
numbers, then divide each by that number.
Step 3 Find a prime number that divides each
quotient, then divide each quotient by that prime.
Continue this process until no prime divides all
remaining quotients.
Step 4 The product of all the prime numbers you
divide by is the GCF.
EXAMPLE 10
Using the Divide by Primes
Method
Use the divide by primes method to find the GCF
of 40, 60, and 100.
SOLUTION
40
60
20
30
10
15
2
3
100
50
25
5
Prime divisor: 2
Prime divisor: 2
Prime divisor: 5
There are no further prime divisors, so
the GCF is 2 x 2 x 5 = 20. This matches our answer from
Example 6.