In this lesson we will review prime and composite numbers. We’ll also review some handy rules that make division faster and easier.

- So far, we’ve covered the various types of numbers (real numbers, rational numbers, irrational numbers, whole numbers, and integers), as well as how they relate to each other using the common device of a number line.
- We’ve also discussed the various properties of addition and multiplication.

2, 3, 5, 7, 11, 13, 17, 19, 23

There is no pattern for finding all of the prime numbers that exist, although mathematicians have found prime numbers with almost eight million digits. This is one of the remaining big math mysteries out there for mathematicians.

*Composite numbers* have more than two factors but not an infinite number of factors. All even numbers (except the number two) are composite, since they can all be divided by two.

Zero is neither prime nor composite. Since any number times zero equals zero, there are an infinite number of factors for a product of zero. A composite number must have a finite number of factors.

One is also neither prime nor composite. The only way to get a product of one is by multiplying 1 x 1. But duplicate factors are only counted once, so one only has one factor. (A prime number has exactly two factors, so one can’t be prime.)

A typical test question would have you identify which number from a list is prime (or composite or neither).

Try this one:

Which of the following is a prime number?

- 33
- 45
- 41
- 51

The correct answer is C. Forty-one is a prime number; the other numbers are composite. Be careful with 51—it comes from 3 x 17.

Divisibility rules are tools to help you do less division. The divisibility rules can help you determine if a number is prime or composite as well as help you break composite numbers into their prime factors, the topic we will cover next.

Divisibility rules are shortcuts to tell you if one number is divisible by another number. The rules cover division by numbers 2–10 and allow you to check numbers up through 100. Click on the numbers in the chart.

Divisibility Rules | ||
---|---|---|

2 | 3 | 4 |

5 | 6 | 7 |

8 | 9 | 10 |

Which of the following numbers is not divisible by 3?

- 4,873
- 7,521
- 12,642
- 25,308

The correct answer is A. The fast way to do this problem is to use the divisibility rule; add the digits of each number looking for the one that is not a multiple of three. The slow way would be to do each division problem—even on a calculator, this method will take longer!

This is another topic you might not have thought about since middle school. The Fundamental Theorem of Arithmetic states that every composite number can be broken down into a unique product of prime numbers.

Prime factorization is the process that finds the prime-number products of a given composite number. Prime factorization is useful in its own right, but many of the applications for it involve fractions, which we’ll cover down the road.

You may be able to find a number’s prime factorization in your head. For example, 12 is 2 x 2 x 3 or 2^{2} x 3. (Prime factorizations are often represented with exponents.)

If you can’t find the prime factorization mentally, try either a Factor-T or a Factor Tree.

*Prime numbers*have exactly two factors, one and itself.*Composite numbers*have more than two factors but not an infinite number.- Zero and one are neither prime nor composite.
*Divisibility rules*are shortcuts that can assist you in determining factors of a number.- Every composite number can be written as a unique product of prime numbers, which is called a number’s
*prime factorization*. - Two handy methods of finding prime factorizations are factor-T’s and factor trees.