# Factors vs. Multiples: What's the Difference?

Edited by Janet White || By Harlon Moss || Updated on October 18, 2023
Factors are numbers that divide another number without leaving a remainder, while multiples are the result of multiplying that number by integers.

## Key Differences

When we discuss factors and multiples, we enter the realm of mathematics, particularly number theory. Factors of a number are those integers which can divide the number exactly, without leaving any remainder. For instance, the factors of 6 are 1, 2, 3, and 6. This is because 6 divided by any of these numbers results in a whole number without any fractions.
Multiples, on the other hand, are numbers derived by multiplying a given number by any integer. Consider the number 3; its multiples are 3, 6, 9, 12, and so on. This list can go on indefinitely, signifying that a number has countless multiples.
Another essential distinction between factors and multiples lies in their quantity. A specific number will only have a limited set of factors, but the list of its multiples is infinite. Take the number 7 as an example. Its only factors are 1 and 7, yet it has an infinite number of multiples: 7, 14, 21, 28, and so on.
The relationship between factors and multiples is bidirectional in some cases. When a number is a factor of another, the second number becomes a multiple of the first. To illustrate, 4 is a factor of 12, and concurrently, 12 is a multiple of 4.
Understanding the distinction between factors and multiples is fundamental in many mathematical operations and problem-solving scenarios. Grasping their concepts helps in tasks like determining the least common multiple or the greatest common divisor, both crucial in fraction-related problems.

## Comparison Chart

### Definition

Numbers that divide another without a remainder.
Result of multiplying a number by different integers.

1 and 5
5, 10, 15, 20...

Limited
Infinite

### Relationship with Number

Integral divisors of the number.
Products when the number is multiplied by integers.

### Range

Typically smaller than or equal to the given number.
Typically larger than the given number.

## Factors and Multiples Definitions

#### Factors

Integers that divide another number precisely.
2 and 3 are factors of 6.

#### Multiples

Several quantities of a set magnitude.
Buying in multiples of ten can sometimes fetch discounts.

#### Factors

Numbers that can multiply together to give the original number.
5 and 4 are factors of 20.

#### Multiples

Numbers that can be divided by a particular number without a remainder.
21 and 42 are multiples of 7.

#### Factors

Integral divisors of a particular number.
1 and 7 are the only factors of 7.

#### Multiples

Repeated occurrences of a particular number.
3 repeated four times gives multiples of 3: 3, 6, 9, and 12.

#### Factors

Quantities contributing to a particular result.
Diet and exercise are factors in overall health.

#### Multiples

Numbers resulting from multiplying a given number by integers.
10, 20, and 30 are multiples of 10.

#### Factors

Elements or circumstances contributing to a result.
Economic factors influence market trends.

#### Multiples

The product of a number and an integer.
14 is a multiple of 7.

#### Factors

One that actively contributes to an accomplishment, result, or process
"Surprise is the greatest factor in war" (Tom Clancy).

#### Multiples

Having, relating to, or consisting of more than one individual, element, part, or other component; manifold.

#### Factors

One who acts for someone else; an agent.

#### Multiples

A number that may be divided by another number with no remainder
4, 6, and 12 are multiples of 2.

#### Multiples

Plural of multiple

## FAQs

#### How are factors of prime numbers identified?

Prime numbers have only two factors: 1 and the number itself.

#### Can zero be considered a multiple of any number?

Yes, zero is a multiple of every number.

#### Are all multiples of a number larger than the number itself?

No, the smallest multiple is the number itself.

#### How are common factors of two numbers determined?

By listing factors of each number and identifying the numbers common to both lists.

#### Which has more variety: factors or multiples?

Multiples, as they are infinite for every number.

#### How are factors and multiples used in real life?

They're utilized in areas like music, design, and financial calculations, among others.

#### Can a number be both a factor and a multiple of another number?

Yes, for any number, 1 and the number itself are both factors and multiples.

#### How can you quickly find multiples of 10?

By appending a zero to any number.

#### How is the greatest common factor of two numbers determined?

By finding the largest number that is a factor of both.

#### Are there negative factors and multiples?

Yes, both factors and multiples can be negative.

#### Can a number have an unlimited number of factors?

No, every number has a limited set of factors.

#### Can two numbers have the same set of factors?

Only if they are the same number.

#### How is the least common multiple of two numbers found?

It's the smallest number that's a multiple of both.

#### Are factors always smaller than or equal to the number?

Yes, factors are always less than or equal to the number.

#### Why are multiples infinite?

Because you can keep multiplying a number by increasingly larger integers.

#### Are the factors of a number always less than its multiples?

Generally, yes, except the number itself, which is both a factor and a multiple.

#### Are 1 and the number itself always factors of the number?

Yes, for every number, 1 and the number itself are factors.

#### Is every number a multiple of 1?

Yes, every number is a multiple of 1.

#### How can technology assist in finding factors and multiples?

Calculators and software can quickly compute factors and multiples for large numbers.

#### Why are factors important in mathematics?

They're crucial for operations like fraction simplification and solving polynomial equations.