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Class Notes for Prime Factorization:
I sometimes require notations and techniques that are not covered completely in the text.
These should always be used in your homework, tests and quizzes.
Prime Factorization is the key to finding GCF’s (Greatest Common Factor) and LCM’s
(Least Common Multiple).
The textbook doesn’t emphasize the importance of doing prime factorization, which is the key skill needed for
all operations involving fractions. The Factor Ladder technique used in the text also works, but it is not very
flexible and it doesn’t use your ability to use shorthand division. The Factor Tree technique exposes the primes
to be multiplied with a minimum of side calculations.
Factor Tree technique for doing Prime Factorization (PF):
1. Write down the number to be PF’ed.
2. Find one prime factor at a time (using tests below), going down one level for each new one.
a. Find all factors of 2 (if it’s an even number).
b. Find all factors of 5 (if it ends on 0 or 5).
c. Find all factors of 3 (if sum of digits divides by 3).
d. Check for other prime factors by using long division.
3. When all lower numbers are prime, write the PF from the “dangling” primes.
Example: Find the PF of 630: (shown as step-by step – circle the primes as you find them)
630
Factor out 2
10
10
10
10
630
630
630
630
ᴧ
15
2 31 5
Factor out 5
ᴧ
2 31 5
ᴧ
15
5
63
ᴧ
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2 31 5
2 31 5
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5 63
5 63
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3 21
3 21
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15
15
Factor out 3
Factor out 3 again
Write the PF as a product of the primes,
Put them in order, Use exponents
630  2  3  3  5  7
 2  32  5  7
3 7
More examples, showing final result.
Find the PF of 144:
Find the PF of 78:
1 44
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2 72
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2 36
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2 18
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2 9
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3 3

Factor out 2
12
18
78
ᴧ
2 39
ᴧ
3 13
Factor out 2


Factor out 2
16
Factor out 2

Factor out 2
78  2  3  13

Factor out 3
the other 3 is prime
All are primes

Factor out 3
13 is also prime

144  2  2  2  2  3  3
 2 4  32
Find the PF of 225:
Find the PF of 46:
25
Factor out 5
Factor out 5
Factor out 3
and other 3
22 5
ᴧ
5 45
ᴧ
5 9
ᴧ
3 3



225  3  3  5  5
 32  5 2
Factor out 2
Can’t be 5 or 3,
7 = 49, can’t be 7
2
46
ᴧ
2 23

46  2  23