Page 205 - Basic College Mathematics with Early Integers

3.2 FACTORS AND SIMPLEST FORM
Objectives

Write a Number as a Product
of Prime Numbers.
Objective Writing a Number as a Product of Prime Numbers
Write a Fraction
To perform operations on fractions, it is necessary to be able to factor a number.
in Simplest Form.
Remember that factoring a number means writing a number as a product. We first
Determine Whether Two practice writing a number as a product of prime numbers.
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Fractions Are Equivalent. Recall from Section 1.6 that since 12 = 2 6, the numbers 2 and 6 are called
factors of 12. A factor is any number that divides a number evenly (with a remain-
Solve Problems by Writing der of 0).
Fractions in Simplest Form. Of all the ways to factor a number, one special way is called the prime factor-
ization. To help us write prime factorizations, we first review prime and composite
numbers.

Prime Numbers
A prime number is a natural number that has exactly two different factors, 1
and itself.

The first several prime numbers are
2, 3, 5, 7, 11, 13, 17

It would be helpful to memorize these.
If a natural number other than 1 is not a prime number, it is called a composite
number.

Composite Numbers
A composite number is any natural number, other than 1, that is not prime.

The natural number 1 is neither prime nor composite.

Now we are ready to define, then find, prime factorizations of numbers.

Prime Factorization
The prime factorization of a number is the factorization in which all the factors
are prime numbers.

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Earlier, we wrote 12 = 2 6 .Although 2 and 6 are factors of 12, the product 2 6
is not the prime factorization of 12 because 6 is not a prime number.
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The prime factorization of 12 is 2 2 3 because
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12 = 2 2 3
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This product is 12 and each
number is a prime number.

Don’t forget that multiplication is commutative, so the order of the factors is not
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important. We can write the factorization 2 2 3 as 2 3 2 or 3 2 2 . Any of
these is called the prime factorization of 12.
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