Topics: Mathematical Language

Mathematical Domain: Number & Operations in Base Ten, Ratios & Proportional Relationships

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

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 × 4, or as 2 × 6, or as 2 × 2 × 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).

- 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, 8 × 1 or 24 × — 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 × 4, or as 2 × 6, or as 2 × 2 × 3. Not all composite numbers can be factored in more than one way, though. For example, 25 can be factored only as 5 × 5.
- The order in which numbers are listed in a factorization does not matter: 3 × 4 and 4 × 3 are the same factorization of 12.

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 factorization of a number is a factorization — a way of expressing that number as a product — consisting only of primes. So, 12 can be expressed as a product many ways — 1 × 2 × 2 × 3, or 3 × 4, 2 × 2 × 3, or 2 × 6 — but only one of those consists solely of primes: 2 × 2 × 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 a prime factorization of 12 must list the 2 twice — 2 × 2 × 3 (or 22 × 3), because 2 × 3, by itself, doesn’t make 12.

Though many numbers can be factored in more than one way, their prime factorization is unique! Apart from order, there is only one way to factor any number into primes!

**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.