# Prime and Composite Numbers Lesson

Problem 1: The area of a rectangular garden is 7 square yards. List all possible whole-number dimensions the garden can have.

Solution: 1 yd x 7 yd

The whole-number dimensions, 1 and 7, of the rectangular garden above, are the factors of the number 7.

Problem 2: The area of a rectangular garden is 8 square yards. List all possible whole-number dimensions the garden can have.

Solution: 1 yd x 8 yd,  2 yd x 4 yd

The whole-number dimensions, 1, 2, 4 and 8, of the rectangular gardens in Problem 2, are the factors of the number 8. In Problem 1, the number 7 has only two factors. The number 7 is prime. In problem 2 above, the number 8 has four factors. The number 8 is composite.

#### Definitions

1. prime number has only two factors: 1 and itself.
2. composite number has more than two factors.
3. The number 1 is neither prime nor composite.

When the area of a rectangle is a prime number, there is only one set of possible dimensions for that rectangle. When the area of a rectangle is a composite number, there are two or more sets of possible dimensions for that rectangle. Each set of dimensions is a pair of factors.

To determine if a number is prime or composite, follow these steps:

1. Find all factors of the number.
2. If the number has only two factors, 1 and itself, then it is prime.
3. If the number has more than two factors, then it is composite.

Example 1: Is the number 2 prime or composite?

Solution: The factors of 2 are 1 x 2. 2 is prime.

Example 2: Is the number 9 prime or composite?

Solution: The factors of 9 are 1 x 9, 3 x 3. 9 is composite.

We have determined if a single number is prime or composite. Let’s look at a range of numbers to see if they are prime or composite. Please note that each range of numbers given in Examples 3, 4 and 5 below are inclusive.

Example 3: Find all prime numbers between 2 and 9.

 factors of 2: 1 x 2 2 is prime factors of 3: 1 x 3 3 is prime factors of 4: 1 x 4, 2 x 2 4 is composite factors of 5: 1 x 5 5 is prime factors of 6: 1 x 6, 2 x 3 6 is composite factors of 7: 1 x 7 7 is prime factors of 8: 1 x 8, 2 x 4 8 is composite factors of 9: 1 x 9, 3 x 3 9 is composite

Solution: The prime numbers between 2 and 9 are 2, 3, 5 and 7.

Example 4: Find all prime numbers between 10 and 19.

 factors of 10: 1 x 10, 2 x 5 10 is composite factors of 11: 1 x 11 11 is prime factors of 12: 1 x 12, 2 x 6, 3 x 4 12 is composite factors of 13: 1 x 13 13 is prime factors of 14: 1 x 14, 2 x 7 14 is composite factors of 15: 1 x 15, 3 x 5 15 is composite factors of 16 1 x 16, 4 x 4 16 is composite factors of 17: 1 x 17 17 is prime factors of 18: 1 x 18, 3 x 6 18 is composite factors of 19: 1 x 19 19 is prime

Solution: The prime numbers between 10 and 19 are 11, 13, 17 and 19.

Example 5: Find all prime numbers between 20 and 29.

 factors of 20: 1 x 20, 2 x 10, 4 x 5 20 is composite factors of 21: 1 x 21, 3 x 7 21 is composite factors of 22: 1 x 22, 2 x 11 22 is composite factors of 23: 1 x 23 23 is prime factors of 24: 1 x 24, 2 x 12, 3 x 8, 4 x 6 24 is composite factors of 25: 1 x 25, 5 x 5 25 is composite factors of 26: 1 x 26, 2 x 13 26 is composite factors of 27: 1 x 27, 3 x 9 27 is composite factors of 28: 1 x 28, 2 x 14, 4 x 7 28 is composite factors of 29: 1 x 29 29 is prime

Solution: The prime numbers between 20 and 29 are 23 and 29.

Example 6: Is the number 31 prime or composite? Explain your answer using full sentences.

Solution 1: The number 31 is prime because its only factors are one and itself.

Solution 2: Thirty-one is a prime number. This is because the number 31 has only two factors: 1 and 31.

Solution 3: I divided the number 31 by all numbers between 1 and 31 and found no factors other than one and thirty-one. Therefore, 31 is prime.

There are many possible ways to explain the solution to this problem. These are just three possible explanations.

Summary: A prime number has only two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite.

The prime numbers between 2 and 31 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 and 31 since each of these numbers has only two factors, itself and 1.

### Exercises

Directions: Read each question below. Select your answer by clicking on its button. Feedback to your answer is provided in the RESULTS BOX. If you make a mistake, choose a different button.

 1 Each of the following numbers is composite EXCEPT: 30, 31, 32, 33, 34, 35, 36, 37, 38, 39 32 and 39 30 and 34 31 and 37 All composite RESULTS BOX:
 2 The prime numbers between 40 and 49 are: 42, 43 and 47 41, 43 and 47  43, 45 and 47 None of the above. RESULTS BOX:
 3 The prime numbers between 50 and 59 are: 53 and 59 51 and 59  53 and 57 None of the above. RESULTS BOX:
 4 The prime numbers between 60 and 69 are: 63 and 69 61 and 67  60 and 65 None of the above. RESULTS BOX:
 5 The prime numbers between 20 and 69 are: 21, 23, 29, 31, 37, 41, 43, 47, 53, 63 and 69 23, 29, 31, 33, 37, 41, 43, 47, 59, 61 and 67 23, 29, 31, 37, 41, 43, 47, 53, 59, 61 and 67 None of the above. RESULTS BOX: