Real Numbers, Prime Numbers, and Absolute Value Study Guide: GED Math

Odd and Even Numbers

An even number is a number that can be divided by the number 2: 2, 4, 6, 8, 10, 12, 14,… An odd number cannot be divided by the number 2: 1, 3, 5, 7, 9, 11, 13,… The even and odd numbers listed are also examples of consecutive even numbers and consecutive odd numbers, because they differ by two.

Here are some helpful rules for how even and odd numbers behave when added or multiplied:

Properties of Real Numbers

Real numbers are all numbers, positive and negative, on the number line:… , –3, –2, –1, 0, 1, 2, 3,… (See the next section for an explanation of number lines.)

The real numbers share properties with which you should be familiar. These properties allow you to change the rules for the order of operations. They can be used to increase speed and accuracy when doing mental math. These properties are also used extensively in algebra when solving equations (for algebra information, see Chapter 6).

Two properties, the commutative and associative properties, deal with expressions that involve a string of all addition operations or a string of all multiplication operations. These properties are for addition and multiplication only.

The commutative property states that when performing a string of addition operations or a string of multiplication operations, the order does not matter: a + b = b + a.

Recall that the order of operations directs you to add or multiply working from left to right.

When you balance your checkbook and have to add up a string of outstanding checks, you can list them all and use the commutative property to arrive at the total by changing the order of addends to add pairs whose ones digits add to 10.

Example

17 + 64 + 35 + 43 + 96 =

Change the order: 17 + 43 + 64 + 96 + 35.

Add 17 and 43 first, because 7 + 3 = 10: 17 + 43 = 60.

The problem becomes 60 + 64 + 96 + 35.

Add 64 and 96 next, because 4 + 6 = 10: 64 + 96 = 160.

The problem becomes 60 + 160 + 35.

Work left to right: 60 + 160 + 35 = 220 + 35 = 255.

In the same way, the commutative property is helpful when multiplying several numbers terms. Change the order to find pairs of numbers whose product would be 10, 100, or 1,000.

Example

4 × 2 × 70 × 50 × 25 =

Change the order to: 4 × 25 × 2 × 50 × 70.

Multiply 4 and 25 first: 4 × 25 = 100.

The problem becomes 100 × 2 × 50 × 70.

Multiply 2 and 50 together: 2 × 50 = 100.

The problem becomes 100 × 100 × 70.

Finish left to right: 10,000 × 70 = 700,000.

The associative property is used when grouping symbols are present. This property states that when you perform a string of addition operations or a string of multiplication operations, you can change the grouping: (a × b) × c = a × (b × c).

Examples

1. 19 + (7 + 16) + 34 =

Change grouping to add 16 and 34 first, because 6 + 4 = 10: 19 + 7 + (16 + 34).

Evaluate the parentheses: 16 + 34 = 50.

The problem becomes 19 + 7 + 50.

Finish, working left to right: 19 + 7 = 26, then 26 + 50 = 76.

2. 15 × (8 × 20) × 5 =

Change grouping to multiply 20 and 5 first, because 20 × 5 = 100: 15 × 8 × (20 × 5).

Evaluate parentheses first: 20 × 5 = 100.

The problem becomes 15 × 8 × 100.

Finish, working left to right: 15 × 8 = 120, then 120 × 100 = 12,000.

The distributive property states that multiplication distributes over addition or subtraction. It deals with two operations—multiplication and addition or multiplication and subtraction. The equation 5(10 + 2) = 5(12) = 60 could also be evaluated as 5(10 + 2) = 5 × 10 + 5 × 2 = 50 + 10 = 60.

The distributive property makes it easy to solve certain math problems quickly.

Example

17 × 5 =

You know that 7 + 10 = 17: 17 × 5 = 5(10 + 7).

Use the distributive property: 5 × 10 + 5 × 7.

Follow the order of operations: 50 + 35 = 85.

Prime and Composite Numbers

A positive integer that is greater than the number 1 is either prime or composite, but not both. A factor is an integer that divides evenly into a number.

• A prime number has only two factors: itself and the number 1.

Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23,…

• A composite number is a number that has more than two factors.

Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16,…

• The number 1 is neither prime nor composite.

Prime factorization is the expression of a positive integer as a product of prime numbers. For example, the prime factor of 72 = 2 × 2 × 2 × 3 × 3.

If the same prime factor occurs more than once, it can be written in exponential form. For instance, the prime factorization of 99 is 3 × 3 × 11. It can also be written as 3² × 11. (See Chapter 5 for a discussion of exponents.)

Prime factorization is usually applied to find the greatest common factor (GCF) or the least common multiple (LCM) of two or more integers.

The greatest common factor (GCF) is the greatest factor that divides into two numbers. To find the GCF of two numbers, list the prime factors of each number. Then, multiply together those factors both numbers have in common. If there are no common prime factors, the GCF is 1.

Example

Find the GCF of 18 and 24.

Prime factors of 18: 2 × 3 × 3

Prime factors of 24: 2 × 2 × 2 × 3

There is one 2 and one 3 in common. The GCF is 2 × 3 = 6.

The least common multiple (LCM) is the smallest number that divides two numbers. To find the LCM of two or more whole numbers, make a list of multiples for each whole number. Continue your lists until at least two multiples are common to all lists. Identify the common multiples. The LCM is the smallest of these common multiples.

Example

Find the least common multiple of 4 and 10.

4: 4, 8, 12, 16, 20, 24,…

10: 10, 20, 30, 40,…

The LCM of 4 and 10 is 20.

Absolute Value

The absolute value of a number or expression is its numerical value without regard to its sign. The absolute value is always positive because it is the distance of a number from zero on a number line. Absolute value is indicated by the symbol | before and after the number or expression.

Example

|–1| = 1

|2 – 4| = |–2| = 2

For order of operations, the absolute value symbol is treated at the same level as parentheses.

Example

5 × |–13 + 3|

First, evaluate the expression inside the absolute value symbol: 5 × |–10|.

Second, evaluate the absolute value: 5 × 10.

Now, perform the multiplication: 5 × 10 = 50.

Practice problems for these concepts can be found at:

Numbers and Operations Practice Problems: GED Math