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# Prime and Composite Numbers

## Only two factors or more than two factors

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Prime and Composite Numbers
License: CC BY-NC 3.0

Simon is working on a riddle.

“I am a number between 1 and 50. The sum of my digits is not prime, but I myself am prime. In a year, you will only see me 7 times. What number am I?”

How can Simon solve the riddle?

In this concept, you will learn how to classify a number as a prime or composite number.

### Classifying Prime and Composite Numbers

Numbers can be classified into two categories. The number of factors that a number has determines whether the number is considered a prime number or a composite number.

Prime numbers are special numbers that are greater than 1 and only have two factors: 1 and the prime number itself. Composite numbers are numbers that have more than two factors. Most numbers are composite numbers.

Here is a prime number.

13

The factors of 13 are only 1 and 13. Therefore, 13 is a prime number. Here is a chart of prime numbers. There are 25 prime numbers between 1 and 100. Take a few minutes to take some notes on prime and composite numbers. Be particularly careful when considering the number “1”. One is neither prime nor composite.

License: CC BY-NC 3.0

Here is a composite number.

44

The factors for 44 is 1, 2, 4, 11, 22, and 44. Therefore, 44 is a composite number; it is made up of more than two factors.

### Examples

#### Example 1

Earlier, you were given a problem about Simon’s riddle.

Use the prime number cart to help solve the riddle. “I am a number between 1 and 50. The sum of my digits is not prime, but I myself am prime. In a year, you will only see me 7 times. What number am I?

First, look at the numbers between 1 an 50 that are prime. There are 15 prime numbers.

\begin{align*}2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47\end{align*}

Then, eliminate the numbers where the sum of the digits is prime.

\begin{align*}\cancel{2}, \cancel{3}, \cancel{5}, \cancel{7}, 11, 13, 17, 19, \cancel{23}, \cancel{29}, 31, 37, \cancel{41}, \cancel{43}, \cancel{47}\end{align*}

Next, find the number you only see 7 times a year. Think about a calendar. 37 is not on a calendar at all. You see 11, 13, 17, 19 more than 7 times in a year.

\begin{align*}\cancel{2}, \cancel{3}, \cancel{5}, \cancel{7}, \cancel{11}, \cancel{13}, \cancel{17}, \cancel{19}, \cancel{23}, \cancel{29}, \mathbf{31}, \cancel{37}, \cancel{41}, \cancel{43}, \cancel{47}\end{align*}

The answer to the riddle is 31.

#### Example 2

Prove that 91 is a prime number.

First, list all the factors of 91. 91 at the very least has the factors 1 and 91. Keep looking for other factors.

Not 2, 3, 4, 5, 6. Let’s divide 91 by 7.

\begin{align*}91 \div 7 = 13\end{align*}

91 also has the factors 7 and 13.

91 is not a prime number. It is a composite number.

#### Example 3

If a number has more than two factors, the number is prime. True or false?

False, numbers with more than two factors are composite numbers.

#### Example 4

The operation associated with factors is addition. True or false?

False, the operation associated with factors is multiplication.

#### Example 5

Why is 29 a prime number?

29 is a prime number because the only two factors for 29 are 1 and 29.

### Review

Identify the following values as prime or composite.

1. 12
2. 10
3. 15
4. 16
5. 56
6. 18
7. 20
8. 22
9. 23
10. 25
11. 27
12. 31
13. 81
14. 48
15. 24
16. 30

To see the Review answers, open this PDF file and look for section 5.3.

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### Vocabulary Language: English

Composite

A number that has more than two factors.

Divisibility Rules

A list of rules which help you to determine if a number is evenly divisible by another number.

Factors

Factors are numbers or values multiplied to equal a product.

Prime

A prime number is a number that has exactly two factors, one and itself.