## Terminology

One area that tends to give many people trouble in mathematics is keeping the terminology straight; as with many disciplines, math has its own language, which is required to be able to communicate. So what are the factor and the product in math?

A factor of an integer is the same as a divisor. It’s another integer that the first number can be divided by such that there is no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. A related concept is a prime factor, which is just what you would expect: a factor that happens to be prime. The prime factors of twelve are two and three.

The product is what you get by multiplying numbers together. We say that the product of two and three is six. Conversely, the quotient is the result of division, so if you divide six by two, you get a quotient of three.

We can express any composite number as the product of its prime factors. This is, unsurprisingly, known as the prime factorization. For example, the prime factorization of twelve is 12 = 2 x 2 x 3; twelve is the product of 2^{2} and 3.

## Why Factor?

Math factors are useful in a variety of applications. For example, finding the prime factorization of two integers allows us to find the lowest common multiple of those numbers, which we can use for things like finding a common denominator so we can compare fractions.

In real life, prime and almost prime numbers arise in a variety of applications. Public key cryptography, for example, uses a trapdoor function – something that's easy to compute in one direction and very hard to computer in another direction. A common function is to take two very large prime numbers and multiply them together. Other people are then given the resulting composite number. Multiplying the two primes is easy, but turning the large number back into its prime factors is very difficult!

## Mnemonics

Having trouble remembering the terms? Think of it this way, putting numbers together produces a larger number, so we call that larger number the product! Note that while this can be a useful memory trick, the factor isn't always larger than both numbers; multiplying 10 by 1/2, for example, will give a product of five.

## The Factor and the Product in Math: Practice Problems

Ready to work with the factor and the product in math problems? See if you can solve the following problems. The answers are below.

What are the factors and prime factors of each of the following numbers?

1) 6

2) 7

What is the product of each set of numbers?

3) 2 and 4

4) 5 and 3 and 2

## Solutions to the Practice Problems

1) The prime factors of six are two and three, which can be multiplied together in only one way (2×3=6). Thus, the factors of six are 1, 2, 3, 6.

2) Seven has no prime factors other than itself – it is a prime number. Thus, its only factors are itself and one.

3) Two times four is eight.

4) Five times three is fifteen, times two is thirty. Alternatively, two times three is six, times five is thirty; we can rearrange the numbers and get the same answer due to the commutative law of multiplication.