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Patterns, Functions, and Algebra
 
Session 6 Part A Part B Homework
 
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Session 6:
Homework

Problem H1

Solution  

Prime numbers have exactly two factors. Now find some numbers that have exactly three factors. What do these numbers have in common? That is, how would you categorize these numbers?


Stop!  Do the above problem before you proceed.  Use the tip text to help you solve the problem if you get stuck.
Look for numbers with three factors, not three prime factors. The number itself and 1 are always factors, so there must be exactly one other factor. When we factor a number, we typically get two distinct factors. How could we get only one new factor?   Close Tip

 

Problem H2

Solution  

There is a way to find the number of factors of a positive integer without writing out all the factors, and it requires finding the prime factorization first. This problem will help you discover that rule.

Go through the table, and list all the factors for each number. Then in the table enter the total number of factors (including the number itself and 1). Look for patterns, and try to write a general rule for the number of factors for any integer.

Integer

Prime Factori-
zation

Number of Factors

2

21

2

4

22

8

23

16

24

2n

2n

3

31

2

9

32

27

33

81

34

3m

3m

6

21 • 31

4

12

22 • 31

18

21 • 32

36

22 • 32

2n • 3m

2n • 3m

show answers

Integer

Prime Factori-
zation

Number of
Factors

2

21

2

4

22

3

8

23

4

16

24

5

2n

2n

(n + 1)

3

31

2

9

32

3

27

33

4

81

34

5

3m

3m

m + 1

6

21 • 31

4

12

22 • 31

6 (i.e., 3 • 2)

18

21 • 32

6 (i.e., 2 • 3)

36

22 • 32

9 (i.e., 3 • 3)

2n • 3m

2n • 3m

(n + 1) • (m + 1)


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Problem H3

Solution  

A number is called a perfect number if the sum of all of its factors is equal to twice the value of the number. What are the two smallest perfect numbers?


 

Problem H4

Solution  

An abundant number is one in which the sum of its factors is greater than twice the number. A deficient number is one in which the sum of its factors is less than twice the number. Which numbers less than 25 are abundant and which are deficient?


 

Problem H5

Solution  

You have seen that every prime number greater than 3 is one less or one more than a multiple of 6. It is also true that every prime number greater than 2 is one more or one less than a multiple of 4. How would you prove this fact?


Next > Session 7: Fractions and Decimals

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