Luttrell 2012
104
3 2 *3 2*2
Name: ______________________
Date: _____
7d – Prime Factorization
Often times it is necessar
y
to break down an item into smaller pieces, whether it be a digestive
s
y
stem, rearranging a postal package contents, troubleshooting a computer problem, or an
y
other
instance. Working with numbers, the
factor
is a number that divides into another evenl
y
. For
e
x
ample, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. A factor tree is a usual algorithm for
finding the factors of a number.
(American) Factor Tree:
(European) Factor Chart
18 - looks like long division
18
2 9 2 9
3 3
3 3 3 1
The usefulness of a factor tree is that
y
ou have divided up the number (18) into its smallest
factors. Those factors of 2 and 3 are onl
y
divisible b
y
1 and itself. That makes those factors
prime
. One is not considered prime; it is
unique
. While the goal of the factor tree is to get
prime factors, the other factors can be found by combining the different primes. The
prime
factorization
is a list of all the prime factors in ascending order. 18 would have a prime
factorization of 2×3×3 or 2×3
2
. The prime factorization of 24 is 2
3
×3 because 24 = 2×12 =
2×2×6 = 2×2×2×3. By combining different prime numbers of 24, you can get the entire factors
of 24: 2, 3, 4, 6, 8, 12, and 24. The factors of 18 are 2, 3, 6, and 9. Factoring can also take a
pol
y
nomial and split it into smaller pol
y
nomials. This idea that pol
y
nomials like
x
2
- 2
x
- 3 can
be factored into
x
- 3 and
x
+ 1 will be developed later.
When adding fractions together,
y
ou need to get a common denominator. Find the Least
Common Multiple (LCM) in order to reduce the work. For 18 and 24 the least common multiple
would be 72. There are two methods for finding the LCM. A common elementar
y
method is to
list the multiples of each number until
y
ou find a common number between the two.
18
36
54
72
90
108
126
144
162
180
198
216
24
48
72
96
120
144
168
192
216
240
264
288
Another method is to use the greatest power of each prime in the prime factorization. The prime
factorization of 18 is 2×3
2
and 24 is 2
3
×3, so the LCM would be 2
3
×3
2
which is 8 ×9 = 72. A
good wa
y
to visualize the result is to use a Venn diagram of the prime factors. Place the prime
factors of 18 in the left circle and the prime factors of 24 in the right circle. The factors in
common should be placed in the shared region. Now take each part of the circles and multiply
the factors together and you get 3∙2∙3∙2∙2 = 72.
Note: GCF is the shared region. This will be explained later.
