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Chapter 1—Whole Numbers

1-10 Factoring

 When you complete the work for this section, you should be able to: State the definition of factors. Identify prime and composite factors of a whole number. Demonstrate your ability to factor whole numbers.

 Definition Any whole number can be expressed in terms of the product (multiplication) of at least two whole numbers. These numbers are the factors of the original whole number. The process of breaking a number down into its factors is called factoring.

Any whole number can be expressed as the product of at least two numbers.  The number 8, for example, can be expressed as 2 • 4:

8 = 2 • 4

The 2 and the 4 are factors of 8.

But the number 8 can be expressed in terms of other factors as well. For instance:

8 = 2 • 2 • 2

And also:

8 = 1 • 8

You can choose any whole number you like, and you can always find at least two factors.

Many whole numbers have just two factors -- the number 1 and the number itself.  Consider 5. The only two whole numbers you can multiply together to equal 5 are 1 and  5.

5 = 1 5

The number 5 is an example of a prime number.

 Definitions A prime number is a whole number that has only one set of factors, itself and 1. A number that is not a prime number is called a composite number.

Examples

Here are some more prime numbers. Take a moment to see if you can find any other combination of numbers, when multiplied together, equal the original.

 2 = 1 • 2 3 = 1 • 3 5 = 1 • 5 7 = 1 • 7 11 = 1 •11 13 = 1 • 13

Here are some composite numbers numbers that are not prime numbers. Notice how each number can be expressed as a product of numbers that do not include 1 and the original number, itself.

 6 = 2 • 3 10 = 2 • 5 12 = 2 • 2• 3 15 = 3 • 5 35 = 5 • 7 27 = 3 • 3• 3

 Notes The ONLY possible factors for a prime number are 1 and the number, itself. Composite numbers can have many different combinations of factors, NOT INCLUDING 1 and the number, itself.

Endless Examples & Exercises

 Determine whether the given number is a prime number or composite number.

Many operations in algebra and higher math require you to factor a number. Further, you are often required to make sure all the factors are, themselves, prime numbers.

 Definitions When the factors of a number are all prime numbers, the factors are said to be the prime factors of that number.

Examples

The prime factors of 20 are 2, 2, and 5

20 = 2 25        All factors are prime numbers

The prime factors of 30 are 2, 3, and 5

30 = 2 • 3 • 5        All factors are prime numbers

The prime factors of 1050 are 2, 3, 5, 5, and 7

1050 = 2 3 5 5 7

Endless Examples & Exercises

 Factor the given number into its prime factors.Continue working these exercises until you can solve the problems without making errors.

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