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.
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Comparison Chart
Definition
Numbers that divide another without a remainder.
Result of multiplying a number by different integers.
Example for number 5
1 and 5
5, 10, 15, 20...
Quantity for a number
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.
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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.
About Author
Written by
Harlon MossHarlon is a seasoned quality moderator and accomplished content writer for Difference Wiki. An alumnus of the prestigious University of California, he earned his degree in Computer Science. Leveraging his academic background, Harlon brings a meticulous and informed perspective to his work, ensuring content accuracy and excellence.
Edited by
Janet WhiteJanet White has been an esteemed writer and blogger for Difference Wiki. Holding a Master's degree in Science and Medical Journalism from the prestigious Boston University, she has consistently demonstrated her expertise and passion for her field. When she's not immersed in her work, Janet relishes her time exercising, delving into a good book, and cherishing moments with friends and family.
