### Your Turn

3.1

1.

Yes. When 54 is divided by 9, the result is 6 with no remainder. Also, 54 can be written as the product of 9 and 6.

3.2

1.

The last digit is 0, so 45,730 is divisible by 5, since the rule states that if the last digit is 0 or 5, the original number is divisible by 5.

3.3

3.4

1.

The last digit is even, so 2 divides 43,568. The sum of the digits is 26. Since 26 is not divisible by 3, neither is 43,568. The rule for divisibility by 6 is that the number be divisible by both 2 and 3. Since 43,568 is not divisible by 3, it is not divisible by 6.

3.5

3.6

1.

The number formed by the last two digits of 43,568 is 68 and 68 is divisible by 4. Since the number formed by the last two digits of 43,568 is divisible by 4, so is 43,568.

3.7

3.8

3.9

3.10

3.11

3.12

3.13

3.14

3.15

3.16

3.17

3.18

3.19

3.20

3.21

3.22

3.23

3.24

3.25

1.

The first person to receive both giveaways would be the person who submits the 11,700th submission.

3.28

3.29

3.30

3.31

3.32

3.33

3.34

1.

−62. Since a larger positive number was subtracted from a smaller positive number, a negative result was expected.

3.35

3.36

3.37

3.38

3.39

3.40

3.41

3.42

3.43

3.44

3.45

3.46

3.47

3.48

3.49

3.50

3.52

3.53

1.

a \times b = 8 \times 26 = 208 and b \times c = 14 \times 12 = 168. The fractions are not equivalent.

3.54

3.55

3.56

3.57

3.58

3.59

3.60

3.61

3.62

3.63

3.64

3.65

3.66

3.67

3.69

3.70

3.71

3.72

1.

Studying math: 5 hours

Studying history: 2.5 hours

Studying writing: 1.25 hours

Studying physics: 1.25 hours

Studying math: 5 hours

Studying history: 2.5 hours

Studying writing: 1.25 hours

Studying physics: 1.25 hours

3.73

3.74

3.75

3.81

3.82

3.83

1.

We want the original price of the item, which is the total. We know the percent, 40, and the percentage of the total, $30. To find the original cost, use \frac{{100\,\times \,{x}}}{{n}}, with x = 30 and n\,{\text{ = }}\,{\text{40}}. Calculating with those values yields \frac{{100\, \times \,30}}{{40}}\, = \,75. So, the original was $75.

3.86

3.87

3.88

3.89

3.90

3.91

1.

The two numbers being subtracted do not have the same irrational part, so the operation cannot be performed without a calculator.

3.95

3.98

3.100

1.

9 \times 8 = 99\. Using that, the problem can be changed to 99 \times 8. Change to 99 = (100 - 1). Using the distributive property, 99 \times 8 = (100 - 1) \times 8 = 100 \times 8 - 1 \times 8 = 800 - 8 = 792.

3.102

3.103

3.104

3.105

3.106

3.107

2.

Since the bases are not the same (one is 3, the other 4), this cannot be simplified using the product rule for exponents.

3.108

3.109

3.110

3.111

3.117

3.

Is not written in scientific notation; The absolute value of –80.91 is not at least 1 and less than 10.

3.119

3.120

3.124

3.125

3.128

3.129

3.130

3.131

3.132

2.

This is not an arithmetic sequence. The difference between terms 1 and 2 is 2, but between terms 3 and 4 the difference is 4. The differences are not the same.

3.

This is an (infinite) arithmetic sequence. Every term is the previous term plus 6. The ellipsis indicates the pattern continues.

3.133

3.134

3.135

3.136

3.137

3.138

2.

It is not a common ratio; term 2 is the first term multiplied by −2, but the sixth term is the fifth term multiplied by 3.

3.141

3.142

3.143

### Check Your Understanding

17.

- 41,{\mkern 1mu} \,\frac{4}{3},\,\,2.75,\,{\mkern 1mu} 0.2\overline {13} are rational; \sqrt {13} is not.

34.

\mathbb{N} \subset \mathbb{Z} | \mathbb{N} \subset \mathbb{Q} | {\mathbb{N}} \subset {\mathbb{R}} | {\mathbb{Z}} \subset {\mathbb{Q}} | {\mathbb{Z}} \subset {\mathbb{R}} | {\mathbb{Q}} \subset {\mathbb{R}} |

52.

No. The difference from term 1 to term 2 is different than the difference from term 4 to term 5.