# Factor

A number that is multiplied by another number to find a product.

Example:

4 x 7 = 28

The factors are 4 and 7.

The terms

factorand multiple are sometimes confused with each other.Factorsof 15 include 3 and 5; multiples of 15 include 30, 45, 60 (and more). See more below and at multiple.

## Meaning

**Factor** can be used as a verb or a noun.

*Verb*: To**factor**a number is to express it as a product of (other)*whole*numbers, called its factors. For example, we can factor 12 as 3 x 4, or as 2 x 6, or as 2 x 2 x 3. So 2, 3, 4, and 6 are all factors of 12.

*Noun*: A**factor**of a number -- let's name that number*N*-- is a number that can be multiplied by something to make*N*as a product. Another way of saying it: a number's factors are divisors of that number; that is, they can divide that number without leaving a remainder.

So, for example, 3 is a factor of 12 because 3 is a counting number and it can be multiplied by 4 to make 12. Again 3 is a factor of 12 because 3 divides 12 without leaving a remainder. The factors of 12 are 1, 2, 3, 4, 6, and 12, because each of those divides 12 without leaving a remainder (or, alternatively, each of those is a counting number that can be multiplied by another counting number to make 12).

### Some finer points

- The
*factors*of a number include the number, itself, and 1. But these are pretty trivial factors, and so when we talk about*factoring*a number, we don't generally include factorizations that include 1 or the number itself. - In the context of numbers, the terms
*factor*(and*multiple*and*divisibility*) are used only in connection with whole numbers. So, for example, even though 12 can be expressed as a product using fractions -- for example, or -- these are not factorizations of 12. - Prime numbers
*have*two factors, themselves and 1, but those are the trivial factors that every number has. Because they cannot be factored in any*other*way, we say that they cannot be factored. For example, 7 "cannot be factored" (even though it*has*the two factors 1 and 7, or could be expressed as a product of non-whole numbers in various ways). - Composite numbers (counting numbers that are neither prime nor 1) can often be factored (expressed as a product of whole numbers) in more than one way. For example, 12 can be factored as 3 x 4, or as 2 x 6, or as 2 x 2 x 3. Not all composite numbers can be factored in more than one way, though. For example, 25 can be factored only as 5 x 5.
- The order in which numbers are listed in a factorization does not matter: 3 x 4 and 4 x 3 are the same factorization of 12.

## More mathematical background

A prime factor of a number is just a factor of that number that is also prime. So, 12 has six factors -- 1, 2, 3, 4, 6, and 12 -- but only two of them (2 and 3) are prime, so it has only two prime factors.

The

prime factorizationof a number is a factorization -- a way of expressing that number as a product -- consistingonlyof primes. So, 12 can be expressed as a product many ways -- 1 x 2 x 2 x 3, or 3 x 4, 2 x 2 x 3, or or 2 x 6 -- but only one of those consists solely of primes: 2 x 2 x 3. (The number 1 is not prime. See prime to learn why.) The numbers 2 and 3 are the only prime factors of 12, but aprime factorizationof 12 must list the 2 twice -- 2 x 2 x 3 (or 2^{2}x 3), because 2 x 3, by itself, doesn't make 12. Though many numbers can befactoredin more than one way, theirprimefactorization is unique! Apart from order, there is onlyoneway to factor any number into primes!

## Related mathematical terms

See multiple, product, and divisor for more.

## What's in a word?

**Factor** is related to *factory*. Just as a factory is a place that *makes* various products, a factor is a number that *makes* other numbers as products. The words *factor* and *factory* come from a Latin root that means "make" or "do." The word *fact* is also related; originally a 'deed,' something that we know is true because it has been done.