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. A prime number has only two factors: 1 and itself.
2. A 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.

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

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

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

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

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.