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Prime Factorization
Math
Prime Factorization Of a Number
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A prime number is a counting number that only
has two factors, itself and one. Counting numbers
which have more than two factors (such as six,
whose factors are 1, 2, 3 and 6), are said to be
composite numbers. When a composite
number is written as a product of all of its prime
factors, we have the prime factorization of the
number.
There are several different methods in which can
be utilized for the prime factorization of a
number.
Using Division
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Prime factors can
be found using
division.
Keep dividing until
you have all prime
numbers. The
prime factors of 78
are 2, 3, 13.
39
2 78
13
3 39
Remember the Divisibility Rules
If the last digit is even, the number is
divisible by 2.
 If the last digit is a 5 or a 0, the number is
divisible by 5.
 If the number ends in 0, it is divisible by
10.
 If the sum of the digits is divisible by 3,
the number is also.
 If the last two digits form a number
divisible by 4, the number is also.
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More divisibility rules…
If the number is divisible by both 3 and 2,
it is also divisible by 6.
 Take the last digit, double it, and subtract
it from the rest of the number; if the
answer is divisible by 7 (including 0), then
the number is also.
 If the last three digits form a number
divisible by 8, then the whole number is
also divisible by 8.
 If the sum of the digits is divisible by 9,
the number is also.
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 Using
the Factor Tree
78
/ \
/
2
/
/
2 x
\
x
39
/
/
3 x
\
\
13
Exponents
72
/
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\
8 x 9
/ \
/ \
2x4x3x3
/ / \ \ \
2 x2 x2x3x3
Another key idea in
writing the prime
factorization of a number
is an understanding of
exponents. An exponent
tells how many times the
base is used as a factor.
72 = 23 x 32
Let’s Try a Factor Tree!
84
/ \
2 x 42
/
/ \
2 x 2 x 21
/
/
/ \
2 x 2 3 x 7
What is the final factorization?
22 x 3 x 7 = 84
Factor Trees do not look the same for the same number,
but the final answer is the same.
72
/ \
8 x 9
/ \
/ \
2x4x3x3
/ \
2 x2 x 2x 3x3
72
/ \
2 x 36
/ /
\
2 x 2 x 18
/ /
/ \
2x 2 x 2 x 9
/
/
/
/ \
2 x 2 x2x 3x 3
Greatest Common Factors
One method to find greatest common
factors is to list the factors of each
number. The largest number is the
greatest common factor.
 Let’s find the factors of 72 and 84.
72
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
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84
1, 2, 3, 4, 6, 12, 14, 21, 28, 42, 84
Prime Factorization is helpful for finding
greatest common factors.
72
/ \
8 x 9
/ \
/ \
2x4x3x3
/ \
2 x2 x 2x 3x3
Take the common prime
factors of each number
and multiply to find the
greatest common factor.
84
/ \
2 x 42
/
/ \
2 x 2 x 21
/
/
/ \
2 x 2
3 x 7
2 x 2 x 3 = 12
Resources
 Brain
Pop – Prime Factors
 Brain Pop - Prime Numbers
 Brain Pop - Exponents
Standards
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Checks for Understanding
0506.2.2 Use the prime factorization of two whole
numbers to determine the greatest common factor and the
least common multiple.
0506.2.2 Use the prime factorization of two whole
numbers to determine the greatest common factor and the
least common multiple.
0506.2.4 Use divisibility rules to factor numbers.
0506.2.10 Use exponential notation to represent repeated
multiplication of whole numbers.
Grade Level Expectations
GLE 0506.2.2 Write natural numbers (to 50) as a product
of prime factors and understand that this is unique (apart
from order).