Factors, Multiples, and Primes, Oh My! What are these strange beasts? By the end of this article, you’ll learn how to find the factors and multiples of a whole number. You’ll also discover prime numbers and what role they play in factorization. These concepts provide important fundamentals for succeeding on the GED Math section.

For more information about what to expect, check out What’s on the GED Math Test?

## Factors

The whole numbers, 1, 2, 3, 4, 5, …, are fascinating! Each one has unique qualities and properties, much like the various chemicals and minerals in our world display different characteristics. And just as physical materials can be broken down into simpler components (elements), so too numbers can be broken down into **factors**.

A **factor** of a number is any number that divides into it with zero remainder. For example, the factors of 6 are: 1, 2, 3, 6. Factors are always less than or equal to the number.

The quickest way to find all factors of a number is to start from the smallest and work your way up. Keep in mind that factors come in pairs. If *a* is a factor of *n*, then there must be another number *b* such that *a* × *b* = *n*. That means that *b* would also be a factor.

Let’s find the factors of 36. I like to build a table with two columns. Start with 1 (which is always a factor), and keep going up, testing each number to see if it’s a factor. Put the factor partner in the other column.

1 | 36 |

2 | 18 |

3 | 12 |

4 | 9 |

6 | 6 |

Notice the bottom line: 6 is a factor, but its factor partner is also 6. This is because 36 is a perfect square (36 = 6^{2}). So the factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36.

Can you find the factors of 120? (The answer will be at the end.)

### Multiples

**Multiples** are numbers that have the original number as a factor. In other words, a multiple of *n* is simply the result of multiplying *n* by another whole number *k* (that is, *n* × *k*).

Multiples are always greater than or equal to the original number. Moreover, there’s an infinite number of them for any given whole number! But don’t worry; there’s a very easy way to generate multiples. Think *multiplication table*!

For example, let’s list the first ten multiples of the number 6.

6 × 1 = | 6 |

6 × 2 = | 12 |

6 × 3 = | 18 |

6 × 4 = | 24 |

6 × 5 = | 30 |

6 × 6 = | 36 |

6 × 7 = | 42 |

6 × 8 = | 48 |

6 × 9 = | 54 |

6 × 10 = | 60 |

The multiples are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, etc.

Now you try it out. Find the first ten multiples of 9.

## Primes

Some numbers have a ton of factors while others don’t seem to have many at all. Any number that has precisely two distinct factors, 1 and itself, is called **prime**. Numbers that have more than two factors are **composite**. The lonely number 1 is neither prime nor composite.

The first ten primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

Primes are important because they are the building blocks of every whole number. All composite numbers have a unique **prime factorization**, that is, a list of the primes or prime powers that are factors of the number. The easiest way to find the prime factorization of a number is to use a **factor tree**.

Starting with 2, the smallest prime, divide into the original number until it can’t be divided further. Then go to the next prime and do the same. Keep going until the last quotient is itself prime.

Let’s see how it works for the number 168.

The numbers on the “leaves” of the tree form the prime factorization. Then we would write:

168 = 2^{3} × 3 × 7.

Now you try your hand at it! Find the prime factorization of 1980.

## Summary

- The factors of a number are numbers that divide evenly into the given number.
- The multiples of a number are numbers found by multiplying the original number by any whole number.
- Primes are numbers with only two distinct factors. They are useful in finding the prime factorization of a number.

### Answers

- Factors of 120 are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120.
- Multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, …
- Prime factorization: 1980 = 2
^{2}× 3^{2}× 5 × 11.