Factors, Multiples, and Divisors
How does the following problem relate to the factors in a product?
Carla has twelve gum balls and wants to share them among friends where each
person receives the same number of gumballs. Carla has several choices for how she shares her gumballs depending on how many friends she shares her gumballs with. What are all the possible ways she can share her gum balls â€"number of people and number of gumballs each receives?
Since each person must receive a whole gumball, the problem is asking for all the possible natural number products that can be formed where the product is twelve. That is,
1 × 12, 2 × 6, 3 × 4, 4 × 3, 6 × 2, and 12 × 1.
The possibilities are:
She keeps all twelve gumballs, 1(12) = 12.
She and a friend each get six gumballs, 2(6) = 12.
She and two friends each get four gumballs, 3(4) = 12.
She and three friends each get three gumballs, 4(3) = 12.
She and five friends each get two gumballs, 6(2) = 12.
She and eleven friends each get one gumball, 12(1) =2.
We may consider the above problem in three different ways: What are all the ways two natural number factors give a product of twelve? What are all the ways we can multiply two natural numbers to get twelve? What are the possible natural number divisors of twelve that give a natural number quotient? These different perspectives for the above problem motivate the concepts of factors, multiples, and divisors.
Definitions for Factors, Multiples, and Divisors: Two numbers are factors of a number if their product is the number. The number is a multiple of a factor. Each factor is a divisor of the number.
General Property when the Natural Numbers is the Universal Set:
a is a factor of b if there is a k so that b = ak with {a, b, k} is a subset of the natural numbers.
b is a multiple of a if there is a number k so that b = ak with {a, b, k} is a subset of the natural numbers.
a is a divisor of b if there is a k so that b = ak with {a, b, k} is a subset of the natural numbers.
Numeric Example:
Since 5 × 8 = 40, both 5 and 8 are factors of 40.
Since 5 × 8 = 40, 40 is a multiple of 5 and 40 is also a multiple of 8.
Since 5 × 8 = 40, both 5 is a divisor of 40 and 8 is also a divisor of 40.
Often we need to find all of the factors or multiples of a number. It is convenient to think of this group of factors as a set.
Example: In the introduction motivation problem, the question was asking for all the natural number factors of twelve. The set of factors of twelve, {1, 2, 3, 4, 6, 12}, is a list of possibilities for the number of people who would receive gumballs.
Example:
The set of all the whole number factors (natural number factors) of 15 is {1, 3, 5, 15}.
The set of all the whole number divisors (natural number factors) of 15 is {1, 3, 5, 15}.
The set of all the natural number multiples of 15 is {15, 30, 45, 60, …, 15n, …}.
The set of all the whole number multiples of 15 is {0, 15, 30, 45, 60, …, 15n, …}.
Note that the universe affects the answer. Zero is a whole number multiple of every number since 0 × a = 0. Also notice that the set of multiples is an infinite set.
Example:
{x : x is a natural number multiple of 4} = {4, 8, 12, 16, 20, 24, …, 4n, …}
{x : x is a whole number multiple of 4} = {0, 4, 8, 12, 16, 20, 24, …, 4n, …}
Example:
{x : x is a natural number factor of 24} = {1, 2, 3, 4, 6, 8, 12, 24}.
{x : x is a whole number factor of 24} = {1, 2, 3, 4, 6, 8, 12, 24}.
{x : x is a natural number divisor of 24} = {1, 2, 3, 4, 6, 8, 12, 24}.
{x : x is a whole number divisor of 24} = {1, 2, 3, 4, 6, 8, 12, 24}.
Note that the set of factors is the same when the universe is either the natural numbers or the whole numbers.
If we are asked for the set of all factors of a value, we must include all the whole number factors for the set to be the correct answer. Notice that the factors generally come in pairs.
However, if the product is a perfect square, such as 6 × 6 = 36, there is only one factor because it would be paired with itself.
Write the set of factors of 18.
Solution
Write the set of whole number multiples of 18.
Solution
Write the set of divisors of 18.
Solution
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.
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, …}.
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}
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
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..."