Solutions for Session5, Part C

See solutions for Problems: C1 | C2

Problem C1

a.

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.

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 3 33: 11 • 3 34: 17 • 2 35: 7 • 5

 Problem C2 Looking at the prime factorization of numbers, you can tell how many factors a number will have in total. For example, the prime factorization of 2 is 21, and 2 has two factors in total, 1 and 2. The prime factorization of 4 is 22, and it has three factors in all: 1, 2, and 4. To further investigate this pattern, let's look at the following: 36 = 32 • 22 will have (2 + 1) • (2 + 1), or nine factors total. 24 = 23 • 31 will have (3 + 1) • (1 + 1), or eight factors total. 81 = 34 will have (4 + 1), or five factors total.