Basic Mathematics for College Students

One method for finding the prime factorization of a number is called a factor
tree.The factor trees shown below are used to find the prime factorization of 90 in
two ways.

1. Factor 90 as 1. Factor 90 as


2. Neither 9 nor 10 are 2. Neither 6 nor 15 are
   prime, so we factor prime, so we factor
   each of them. each of them.


3. The process is 3. The process is
   complete when complete when
   only prime numbers appear only prime numbers appear
   at the bottom of all branches. at the bottom of all branches.

Either way, the prime factorization of 90 contains one factor of 2, two factors of 3,
and one factor of 5. Writing the factors in order, from least to greatest, the prime-
factored formof 90 is It is true that no other combination of prime
factors will produce 90. This example illustrates an important fact about composite
numbers.

2  3  3 5.

9 10. 6 15.

1.7 Prime Factors and Exponents 83

Prime Factorization

To find the prime factorizationof a whole number means to write it as the

product of only prime numbers.

Fundamental Theorem of Arithmetic

Any composite number has exactly one set of prime factors.

EXAMPLE (^5) Use a factor tree to find the prime factorization of 210.
StrategyWe will factor each number that we encounter as a product of two
whole numbers (other than 1 and itself) until all the factors involved are prime.
WHYThe prime factorization of a whole number contains only prime numbers.
Solution
Self Check 5
Use a factor tree to find the
prime factorization of 126.
Now TryProblems 61 and 71
Factor 210 as (The resulting prime factorization will be the same
no matter which two factors of 210 you begin with.) Since 7 is prime, circle
it. That branch of the tree is completed.
Since 30 is not prime, factor it as (The resulting prime
factorization will be the same no matter which two factors of 30 you
use.) Since 5 is prime, circle it. That branch of the tree is completed.
Since 6 is not prime, factor it as Since 2 and 3 are prime, circle
them. All the branches of the tree are now completed.
2 3.
5 6.
7 30.
The prime factorization of 210 is Writing the prime factors in
order, from least to greatest, we have
Check: Multiply the prime factors. The product should be 210.
Write the multiplication in horizontal form.
Working left to right, multiply 2 and 3.
Working left to right, multiply 6 and 5.
 210 Multiply 30 and 7. The result checks.

 30  7

2  3  5  7  6  5  7

210  2  3  5 7.

7  5  2 3.

90

910

3 3 2 5

90

615

2 3 3 5

210

730

56

23