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
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.
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
Interactive Math Goodies Software
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