Prime Factorization
A composite number can be expressed as a product of prime numbers. This product is called the
prime factorization of that number.
e.g. Factor 72 as a product of prime numbers.
Solutions:
a)
Use a factor tree:
72
8
2
x
x
4
2
x
9
3
x
3
2
!72 = 23 " 32
To create a Factor Tree:
Find any two factors of the number (do not use 1 as a factor). Continue to factor any
factor that is a composite number until all factors are primes.
b)
Use division:
2) 72
2) 36
2)18
3) 9
3) 3
1
!72 = 23 " 32
To use the Division Method:
Divide by any prime factor. (note: the quotient is written below the corresponding
dividend) Continue to divide the resulting quotient by prime factors that divide exactly
unitl the quotient is 1.
Exercises:
1.
Express each of the following numbers as a product of prime numbers using a factor tree.
Express your answer in exponential form:
a)
20
b)
64
c)
80
d)
48
e)
100
f)
2.
3.
4.
144
g)
960
Factor each of the following numbers as a product of prime numbers:
a)
14
b)
54
c)
72
d)
135
e)
168
f)
600
g)
306
h)
36
i)
75
j)
135
k)
147
l)
200
m)
148
n)
800
o)
201
a)
is 6 a factor of 24?
b)
is 5 a factor of 40?
c)
is 9 a factor of 56?
d)
is 18 a factor of 90?
What is the second factor when:
a)
7 is one factor of 21?
b)
8 is one factor of 72?
c)
d)
1 is one factor of 217?
14 is one factor of 98?
5.
Name all of the prime factors of:
a)
48
b)
132
c)
225
d)
400
e)
12155
6.
Name all of the factors of:
a)
48
b)
132
c)
225
d)
400
e)
12155
7.
8.
Name the common factors of each pair of numbers:
a) 12 and 16
b) 63 and 84
c) 32 and 104
d) 105 and 150
Reduce the following fractions:
a)
!
12
16
b)
!
63
84
c)
!
32
104
d)
!
105
150
9.
Name the common factors of each group of numbers:
a) 24, 60, 96
b) 39, 65, 91
c) 36, 81, 108
10.
Kathy found that the prime factors of two numbers were:
d) 252, 588, 420
2, 5, 3, 13, and 2, 7, 13, 11.
What is the greatest common factor of these two numbers?
11.
Find the highest common factor of 20, 36, and 60.
12.
Find the smallest value of n which will make 12 * n divisible by 28.
13.
What is the smallest value of n which will make 30 * n a perfect square?
How about a perfect cube?
14.
The four digit number 8MN9 is a perfect square. What is the value of M + N?
15.
Challenge
The city of Scarborough organized a hockey league. However,
the league organizer discovered that when the players were
divided into teams of 6, teams of 7 or teams of 8, there was
always one player left over. Finally, another player joined the
league and now the players could be divided into teams of 10
without any left overs. How many teams were in the league?