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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.

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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.

Definitionsprime 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 22 is prime
factors of 3:1 x 33 is prime
factors of 4:1 x 4, 2 x 24 is composite
factors of 5:1 x 55 is prime
factors of 6:1 x 6, 2 x 36 is composite
factors of 7:1 x 77 is prime
factors of 8:1 x 8, 2 x 48 is composite
factors of 9:1 x 9, 3 x 39 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 510 is composite
factors of 11:1 x 1111 is prime
factors of 12:1 x 12, 2 x 6, 3 x 412 is composite
factors of 13:1 x 1313 is prime
factors of 14:1 x 14, 2 x 714 is composite
factors of 15:1 x 15, 3 x 515 is composite
factors of 161 x 16, 4 x 416 is composite
factors of 17:1 x 1717 is prime
factors of 18:1 x 18, 3 x 618 is composite
factors of 19:1 x 1919 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 520 is composite
factors of 21:1 x 21, 3 x 721 is composite
factors of 22:1 x 22, 2 x 1122 is composite
factors of 23:1 x 2323 is prime
factors of 24:1 x 24, 2 x 12, 3 x 8, 4 x 624 is composite
factors of 25:1 x 25, 5 x 525 is composite
factors of 26:1 x 26, 2 x 1326 is composite
factors of 27:1 x 27, 3 x 927 is composite
factors of 28:1 x 28, 2 x 14, 4 x 728 is composite
factors of 29:1 x 2929 is prime

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

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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, 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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