Factors, Multiples, and Divisors

More on Divisors

Since multiplication and division are inverse operations, the natural number divisors of a value are the same as the factors of that value. It may seem confusing to have two different names for the same set of values, but in some contexts (multiplying contexts) it makes sense to call these values the set of factors, while in other contexts (dividing contexts) it makes sense to call these values divisors.

Even and Odd Numbers

A natural number ( or whole number) is an even number if it is a multiple of two. A natural number (or whole number) that is not an even number is an odd number.

General Property:

A value of the form 2n, where n is a counting number ( or a whole number), is an even number.
A value in the form of 2n – 1 where n is a counting number is an odd number.
A value in the form of 2n + 1 where n is a whole number is an odd number.

Note that an odd number is always one less (or one more) than some even number, 2n.

Set-Builder Notation:

The set of even counting numbers is {x : x = 2n where n ε N }.
The set of odd counting numbers is {x : x = 2n – 1 where n ε N }.
The set of even whole numbers is {x : x = 2n where n ε W }.
The set of odd whole numbers is {x : x = 2n + 1 where n ε W }.

Roster Notation:

The set of even counting numbers is {2, 4, 6, 8, 10, …}.
The set of odd counting numbers is {1, 3, 5, 7, 9, …} .
The set of even whole numbers is {0, 2, 4, 6, 8, 10, …}.
The set of odd whole numbers is {1, 3, 5, 7, 9, …}.

Some Other Factor Facts

• A counting number that ends in an even digit is an even number.

• A counting number that ends in the digit 5 or 0 has 5 as a factor.

•  A counting number that ends in the digit 0 has 10 as a factor.

• A counting number that ends in two zeros has 100 is a factor.

Examples with Sets

When working with sets, mathematical or corresponds to the set operation union.

Example:

{x : x is a factor of 12 or x is a factor of 10}

= {x : x is a factor of 12} U {x : x is a factor of 10}

= {1, 2, 3, 4, 6, 12} U {1, 2, 5, 10}

= {1, 2, 3 ,4, 5, 6, 10, 12}

When working with sets, mathematical and corresponds to the set operation intersection.

Example:

{x : x is a factor of 12 and x is a factor of 10}

= {x : x is an factor of 12} ∩  {x : x is a factor of 10}

= {1, 2, 3, 4, 6, 12} ∩ {1, 2, 5, 10}

=   {1, 2}

 Self Check Problem

 Find the roster form for each of the following sets:

{x : x is an odd factor of 18 or x is an even divisor of 20}

Solution

{x : x is an even factor of 18 and x is an even multiple of 3}

Solution 

Joke or Quote

Teacher: "Divide fourteen sugar cubes into three cups of coffee so that each cup has an odd number of sugar cubes in it."
Student: "That's easy: one, one, and twelve."
Teacher: "But twelve isn't odd!"
Student: "Twelve cubes is an odd number of cubes to put in a cup of coffee..."

return to top | previous page