Session 5, Part C:
Factors

 You saw that you could devise tests for such numbers as 6 and 18 based on their relative prime factors. Let's explore factors further. A prime number is a number with exactly two factors. For example, the number 1 is not a prime number because it only has one factor, 1. The number 3 is a prime number because it has exactly two factors, 1 and 3. Note 4 Some numbers factor into two factors only, while others may have two factors, one or both of which can be factored further. Note 5 An important distinction can be made between the terms "factor" and "prime factor." By factors, we mean all the factors of a number. To find all the factors of 12, you can list them as shown below: 1 • 12 2 • 6 3 • 4 You know you can stop here because the next factor on the left would be 4, and you already have it listed on the right. To find the prime factors of 12, you could use a factor tree: The numbers on the bottom branch of this tree are the prime factors of 12 -- they can't be factored any further. So we say that 12 has only two prime factors, 2 and 3, and the prime factorization of 12 is 22 • 3. Note that we could have started the factor tree with the factors 3 and 4, and we would have derived the same prime factorization, 22 • 3.

Problem C1

Complete the following tables to explore the factors and prime factorization of the numbers from 2 to 36.

2 - 36

 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

 a. What are the factors of the numbers from 2 to 36? Enter each number in the appropriate column in the table below. For example, enter 4: 1, 2, 4 in the Three Factors column and 16: 1, 2, 4, 8, 16 in the Five Factors column.

Prime Numbers: Two Factors

Three Factors

Four Factors

Five Factors

Six or More Factors

4: 1, 2, 4

 16: 1, 2, 3, 4, 8, 16
 12: 1, 2, 3, 4, 6, 12

Prime Numbers: Two Factors

Three Factors

Four Factors

Five Factors

Six or More Factors

2: 1, 2

4: 1, 2, 4

6: 1, 2, 3, 6

16: 1, 2, 4, 8, 16

 12: 1, 2, 3, 4, 6, 12

3: 1, 3

9: 1, 3, 9

8: 1, 2, 4, 8

 18: 1, 2, 3, 6, 9, 18

5: 1, 5

25: 1, 5, 25

10: 1, 2, 5, 10

 20: 1, 2, 4, 5, 10, 20

7: 1, 7

14: 1, 2, 7, 14

 24: 1, 2, 3, 4, 6, 8, 12, 24

11: 1, 11

15: 1, 3, 5, 15

 28: 1, 2, 4, 7, 14, 28

13: 1, 13

21: 1, 3, 7, 21

 30: 1, 2, 3, 5, 6, 10, 15, 30

17: 1, 17

22: 1, 2, 11, 22

 32: 1, 2, 4, 8, 16, 32

19: 1, 19

26: 1, 2, 13, 26

 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

23: 1, 23

27: 1, 3, 9, 27

29: 1, 29

33: 1, 3, 11, 33

31: 1, 31

34: 1, 2, 17, 34

35: 1, 5, 7, 35

 b. What are the prime factors of the numbers from 2 to 36? Enter each in the appropriate column in the table below. For example, enter 4: 22 in the Two Prime Factors column and 16: 24 in the Four Prime Factors column.

Prime Numbers

Two Prime Factors

Three Prime Factors

Four Prime Factors

Five Prime Factors

 2 4: 22 16: 24 12: 3 • 22

Prime Numbers

Two Prime Factors

Three Prime Factors

Four Prime Factors

Five Prime Factors

 2 4: 22 8: 23 16: 24 32: 25 3 6: 3 • 2 12: 3 • 22 24: 3 • 23 5 9: 32 18: 32 • 2 36: 32 • 22 7 10: 5 • 2 20: 5 • 22 11 14: 7 • 2 27: 33 13 15: 5 • 3 28: 7 • 22 17 21: 7 • 3 30: 5 • 3 • 2 19 22: 11 • 2 23 25: 52 29 26: 13 • 2 31 33: 11 • 3 34: 17 • 2 35: 7 • 5

The Fundamental Theorem of Arithmetic states that every positive integer other than 1 has a unique factorization into primes (up to rearrangement of the factors). Now, any negative integer is simply ­1 times a positive integer. So we can extend the theorem to all integers in a natural way: each integer (except, of course 1, 0, and ­1) can be written uniquely as a product of primes and either +1 or ­1.

Here are some unique prime factorizations:

12 = 1 • 22 • 3
(Note that we usually leave off the 1 for positive numbers.)

-12 = -1 • 22 • 3

36 = 32 • 22 will have (2 + 1) • (2 + 1), or 9 factors total.

 Problem C2 Look at the prime factorizations of numbers. Do you see any patterns? For example, how many factors in totalŠ will have based on its prime factorization?

Make a table that allows you to complete the prime factorization and total number of factors. Using numbers that are powers of 2 may be a useful way to see that pattern initially.   Close Tip

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