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How Many Factors Does a Number Have?
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1
Factors
Let a and b be positive integers.
Definition 1.1 — factor. a is a factor of b if there exists a positive integer k such that ak = b.
Some books use the term divisor instead of factor. Factors and divisors are the same thing.
Example 1.1 If a =  and b =  then  is a factor of  because there exists a positive integer
k =  such that () = .
Example 1.2 If a =  and b =  then  is a factor of  because there exists a positive
integer k =  such that () = .
Example 1.3 If a =  and b =  then  is a factor of  because there exists a positive integer
k =  such that () = .
Every positive integer n has at least two factors:  and n.
Example 1.4 If a =  and b =  then  is not a factor of  because there (k) =  implies k =  .
So there doesn’t exist a positive integer k such that (k) = .
Problems:
1.  is a factor of . What is k? Find another factor of .
2.  is a factor of . What is k? Find another factor of .
3.  is a factor of . What is k? Find another factor of .
4.  is a factor of . What is k? Can you find another factor of ?
5.  is not a factor of . What is k?
6.  is not a factor of . What is k?
7.  is not a factor of . What is k?
If a number n is not a square number then each factor pairs with another factor. The two
factors a and b have the property that ab = n.
Definition 1.2 — prime. A positive integer p which is greater than  is prime if the only
factors are  and p.
Example 1.5 The number  is prime because the only factors of  are  and .
Example 1.6 The number  is prime because the only factors of  are  and .
The first  prime numbers are , , , , , , , , , .
2 Determine the Number of Factors of a Number
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To determine whether a number n is prime, it suffices to show that no prime less than or equal
√
to n is a factor of n.
√
 < . The prime numbers less than
 are , , , . Since () =  the number  is not prime.
Example 1.7 To determine if  is prime, observe that
Definition 1.3 — composite. A positive integer m which is greater than  is composite if
m = ab for some positive integers a and b which are greater than .
So the number  is composite. If you pay close attention to the definitions you’ll find that  is
neither prime nor composite. Intuitively, every number can be broken down into prime factors in
much the same way that matter is broken down into atoms. The number  is unimportant from
this perspective. Moreover, if  is not considered then the Fundamental Theorem of Arithmetic
asserts that there is only one way to break down a number into its prime factors if the order of
the factors is unimportant.
Theorem 1.1 — Fundamental Theorem of Arithmetic. Every positive integer greater than 
can be written as the product of prime numbers in exactly one way.
The order of the numbers (which are factors) is unimportant. It’s convention to write the
primes from smallest to largest.
Example 1.8 The number  can be written as ()
Example 1.9 The number  can be written as  .
Example 1.10 The number  can be written as   .
2
Determine the Number of Factors of a Number
To determine the number of factors of an integer, n, find the numbers from  to n which are
factors of n.
Example 2.1 Determine the factors of . From the integers , , ,  the factors are , , . So
there are  factors.
Example 2.2 Determine the factors of . From the integers , , . . . ,  the only factors are 
and . So there are  factors.
Example 2.3 Determine the factors of . From the integers , , . . . ,  the only factors are , , ,
. So there are  factors.
Problems:
1. How many factors does  have?
2. How many factors does  have?
3. How many factors does  have?
4. How many factors does  have?
List them.
List them.
List them.
List them.
The Fundamental Principle of Counting (Multiplication Rule) gives us an easy way to determine
the number of factors that a number has. Remember this theorem:
3 Summary
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Theorem 2.1 — Fundamental Theorem of Counting. If there are n tasks that need to be done
and Task  can be done in t ways, Task  can be done in t ways, . . . , Task n can be done in
tn ways then Task  followed by Task  followed by . . . followed by Task n can be done in
tt · · ·tn ways.
Every factor can be determined by combining the Fundamental Theorem of Arithmetic and the
Fundamental Theorem of Counting. After finding the prime factorization of a number, divide
the factors into two. The first set of factors will be the factor for the number. The second set of
factors will be the value of k.
Example 2.4 Determine the number of factors of . First, determine the prime factorization
of  =  . One way to divide the factors is  and  . This would correspond to ( ) = 
which simplifies to () = . So  is a factor because it goes in k =  times. Another way
to divide the factors is  and (). This would give  (()) =  or just () = . So  is a
factor and k = . Likewise, dividing the factorization as  and  corresponds to the factor  and
k =  giving () = . There is also  and  giving the factor  and k =  from which we see
() = . This is where the Fundamental Theorem of Counting comes in. To determine the
factors of  we have Task : choose the exponent of . This can be done in  ways: an exponent
of , , , or . Task : choose the exponent of . This can be done in  ways: an exponent of 
or . Therefore, by the Fundamental Theorem of Counting there are () =  possible factors.
With a little more work you can find the factors are , , , , , , , .
Example 2.5 Determine the number of factors of . Since  = ()()() we have Task :
choose the exponent of . There are two choices ( or ). Task : choose the exponent of .
There are two choices ( or ). Task : choose the exponent of . There are two choices ( or
). By the Fundamental Theorem of Counting there are ()()() =  different factors. These
factors are , , , , , , , .
Problems:
1. How many factors does  have? How many can you find?
2. How many factors does  have? How many can you find?
3. How many factors does  have? How many can you find?
4. How many factors does  have? How many can you find?
5. How many factors does  have? How many can you find?
6. How many factors does () have? How many can you find?
7. How many factors does () have? How many can you find?
8. How many factors does () have? How many can you find?
9. How many factors does () have? How many can you find?
10. How many factors does () have? How many can you find?
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Summary
Make sure you know
1. what a factor of a positive integer is.
2. what a prime number is.
3. how to find the prime factorization of a number.
4. how to determine the number of factors a number has.
5. how to state the Fundamental Theorem of Counting.
4 Answers
4
Answers
Problems:
1.
2.
3.
4.
5.
6.
7.
k =  so  is another factor of 
k =  so  is another factor of 
k =  so  is another factor of 
k = . No other number pairs with  but  will always be a factor.

k = 

k =  = 
k = 

Problems:
1.
2.
3.
4.
 has  factors: , , , 
 has  factors: , , , , 
 has  factors: , , , , , 
 has  factors: , , , , , , , , , , , 
Problems:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
 =   will have ( + )( + ) =  factors
 =  () will have ( + )( + )( + ) =  factors
 is prime so it has  factors: , 
 =  ·  ·  has ( + )( + )( + ) =  factors
 =  ·  has ( + )( + ) =  factors
() =  ·  has ( + )( + ) =  factors
() =  ·  has ( + )( + ) =  factors
() =  ·  has ( + )( + ) = ,  factors
() =  ·  has ( + )( + ) = ,  factors
() =  ·  has ( + )( + ) = ,  factors
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