Primes and Composites
Definitions: A Prime Number is a counting (natural) number with exactly two different factors.
A natural number with more than two factors is called a Composite Number.
Fundamental Theorem of Arithmetic: Each composite number can be expressed as the
product of primes in exactly one way (except for the order of the factors).
In other words, every natural number larger than one is prime, or the product of primes.
Example: 36 is a composite number. It can be written as a product of primes as follows:
2  2  3  3 ; or to write it in a shorter way, 2 2  3 2
Definition: If a and b are whole numbers b  0 , and there is a whole number q such that
a  b  q , then we say:
a) b divides a b | a 
b) b is a factor of a
c) b is a divisor of a
d) a is a multiple of b
Question: What are the factors of 48?
Question: What are some multiples of 6?
Question: Is 573 prime?
Prime Number Test: We can check if n is prime by checking divisibility of all primes less than
the square root of n. Here’s why:
If a number n is not a prime, it can be factored into two factors a and b: n = a×b
If both a and b were greater than the square root of n, a×b would be greater than n. So at
least one of those factors must be less or equal to the square root of n, and to check if n is
prime, we only need to test for factors less than or equal to the square root.
©Dr Barbara Boschmans/Dr Brian Beaudrie
Page 1 of 4
Primes and Factoring
The table below includes a partial list of the factors and the sum of those factors for the
numbers 1 – 50. Fill in the rest of the table, then use the table to answer the questions.
Numbers
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Factors
1
1, 2
1, 3
1, 2, 4
1, 5
1, 2, 3, 6
1, 7
1, 2, 4, 8
1, 3, 9
1, 2, 5, 10
1, 11
1, 2, 3, 4, 6, 12
1, 13
1, 2, 7, 14
1, 3, 5, 15
1, 2, 4, 8, 16
1, 17
1, 2, 3, 6, 9, 18
1, 19
1, 2, 4, 5, 10, 20
1, 2, 11, 22
1, 23
Number of Factors
1
2
2
3
2
4
2
4
3
4
2
6
2
4
4
5
2
6
2
6
4
2
8
Sum of the Factors
1
3
4
7
6
12
8
15
13
18
12
28
14
24
24
31
18
39
20
42
32
60
1, 2, 13, 26
4
1, 2, 4, 7, 14, 28
1, 29
6
56
8
2
32
1, 31
1, 2, 4, 8, 16, 32
48
1, 47
4
4
4
9
2
4
4
8
2
8
2
6
6
4
2
1, 7, 49
1, 2, 5, 10, 25, 50
3
6
57
93
1, 37
1, 2, 19, 38
1, 3, 13, 39
1, 2, 4, 5, 8, 10, 20, 40
1, 41
1, 2, 3, 6, 7, 14, 21, 42
1, 43
©Dr Barbara Boschmans/Dr Brian Beaudrie
48
91
38
90
42
44
48
Page 2 of 4
1. List the numbers that have exactly two factors:
a. What kinds of numbers have exactly two factors?
b. What are the two factors?
c. Predict the next number that will have exactly two factors:
2. List the numbers that have exactly three factors:
a. What kinds of numbers have exactly three factors?
b. What are the three factors?
c. Predict the next number that will have exactly three factors:
3. The numbers below have exactly four factors. They have been separated into two
sets, A and B.
A = {6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46}
B = {8, 27}
a. What is the next number that belongs to the number in Set A? ________
b. What is the next number that belongs to the numbers in Set B? ________
c. How are the numbers in Set A alike?
d. What kinds of numbers are in Set B?
©Dr Barbara Boschmans/Dr Brian Beaudrie
Page 3 of 4
4. List the numbers in the table that have exactly five factors:
a. What kinds of numbers have exactly five factors?
b. Predict the next number that will have exactly five factors:
5. From the table you can see that the number 4 has three factors, 12 has six factors, and
15 has four factors. There is a formula for determining the number of factors (without
writing them out) for any given number. It is based on the prime factorization of that
number. Complete, then use the table below to help you discover the formula:
Number
Prime
Factorization
4
8
10
12
18
20
24
36
40
22
23
21  51
22  31
Number
of
factors
3
4
4
6
Formula (describe it in words):
6. Use the formula to determine the number of factors of 64, 72, and 100.
7. What is the sum of the factors of a counting number n if:
a. n is a prime number? ____________________
b. n has exactly three factors? ___________________
©Dr Barbara Boschmans/Dr Brian Beaudrie
Page 4 of 4