Motivation Problem

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Factors, Multiples, and Divisors

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More on Divisors

# 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.

## Self Check Problems

Write the set of factors of 18.

Write the set of whole number multiples of 18.

Write the set of divisors of 18.