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

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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**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.

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:

Find all factors of the number.If the number has only two factors, 1 and itself, then it is prime.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.

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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, 3932 and 3930 and 3431 and 37All composite
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2. |
The prime numbers between 40 and 49 are:42, 43 and 4741, 43 and 47 43, 45 and 47None of the above.
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3. |
The prime numbers between 50 and 59 are:53 and 5951 and 59 53 and 57None of the above.
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4. |
The prime numbers between 60 and 69 are:63 and 6961 and 67 60 and 65None of the above.
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5.
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The prime numbers between 20 and 69 are:21, 23, 29, 31, 37, 41, 43, 47, 53, 63 and 6923, 29, 31, 33, 37, 41, 43, 47, 59, 61 and 6723, 29, 31, 37, 41, 43, 47, 53, 59, 61 and 67None of the above.
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