Question

1.Show that if d ≠ 0, then d | (-a) and -d | a.
2.Show that it is false that a>b implies a|b.
3.Is 980637 divisible by 7? Show.
4.Determine whether of the following are divisible by 3, 5, 7, 9, or 11 using the methods described int he text:
A. 1969
B. 28350
C. 1421
D. 17303
E. 116424
F. 1089
5.Classify each of the following as true or false:
A. 6 is a divisor of 24.
B. 40 is a multiple of 8.
C. 0 divides 10.
D. 13 is a factor of 33.
E. 12 divides 6.
6.Show that 23n -1 is divisible by 7.
7.Show that 5n - 1 is divisible by 4.

Answer 1

1. Show that if $d \neq 0$, then $d \mid (-a)$ and $-d \mid a$.

Solution.

For example:

$$a = 10; d = 5$$ $$\frac{-10}{5} = -2; \frac{10}{-5} = -2$$

2. Show that it is false that $a > b$ implies $a \mid b$.

Solution.

Let

$$a = b + k; k - \text{positive number}$$

Then

$$\frac{b}{a} = \frac{b}{b + k} < 1$$

3. Is 980637 divisible by 7? Show.

Solution.

$$980 - 637 = 343$$ $$34 - 2 \cdot 3 = 28$$

So 980637 is divisible by 7.

4. Determine whether of the following are divisible by 3, 5, 7, 9, or 11 using the methods described in the text:

A. 1969

Solution.

$$1 + 9 + 6 + 9 = 25$$

It is not divisible by 3 or 9.

$$\text{last digit } 9 \neq 0 \text{ or } 5$$

It is not divisible by 5.

$$1 + 6 \neq 9 + 9$$

It is not divisible by 11.

$$196 - 2 \cdot 9 = 178$$ $$17 - 2 \cdot 8 = 1$$

It is not divisible by 7.

B. 28350

Solution.

$$2 + 8 + 3 + 5 = 17$$

It is not divisible by 3 or 9.

$$\text{last digit } 0$$

It is divisible by 5.

$$2 + 3 \neq 8 + 5$$

It is not divisible by 11.

$$2835 - 2 \cdot 0 = 2835$$ $$283 - 2 \cdot 5 = 273$$ $$27 - 2 \cdot 3 = 21$$

It is divisible by 7.

C. 1421

**Solution.**

$$1 + 4 + 2 + 1 = 8$$

It is not divisible by 3 or 9.

$$\text{last digit } 1 \neq 0 \text{ or } 5$$

It is not divisible by 5.

$$1 + 2 \neq 4 + 1$$

It is not divisible by 11.

$$142 - 2 \cdot 1 = 140$$

It is divisible by 7.

D. 17303

**Solution.**

$$1 + 7 + 3 + 3 = 14$$

It is not divisible by 3 or 9.

$$\text{last digit } 3 \neq 0 \text{ or } 5$$

It is not divisible by 5.

1 + 3 + 3 = 7

It is divisible by 11.

1730 - 2 · 3 = 1724

172 - 2 · 4 = 164

16 - 2 · 4 = 8

It is not divisible by 7.

E. 116424

Solution.

1 + 1 + 6 + 4 + 2 + 4 = 18

It is divisible by 3 and 9.

last digit 4 ≠ 0 or 5

It is not divisible by 5.

1 + 6 + 2 = 1 + 4 + 4

It is divisible by 11.

424 - 116 = 308 ; 30 - 2 · 8 = 14

It is divisible by 7.

F. 1089

Solution.

$$1 + 8 + 9 = 18$$

It is divisible by 3 and 9.

$$\text{last digit } 9 \neq 0 \text{ or } 5$$

It is not divisible by 5.

$$1 + 8 = 9$$

It is divisible by 11.

$$108 - 2 \cdot 9 = 90$$

It is not divisible by 7.

5. Classify each of the following as true or false:

A. 6 is a divisor of 24.

Solution.

$$\frac{24}{6} = 4 \text{ ; true}$$

B. 40 is a multiple of 8.

Solution.

$$\frac{40}{8} = 5 \text{ ; true}$$

C. 0 divides 10.

Solution.

$$\frac{10}{0};\text{ false}$$

D. 13 is a factor of 33.

Solution.

$$\frac{33}{13} = 2 + \frac{7}{13};\text{ false}$$

E. 12 divides 6.

Solution.

$$\frac{6}{12} = \frac{1}{2};\text{ false}$$

6. Show that $23n - 1$ is divisible by 7.

Solution.

$23n - 1$ is not always divisible by 7

For example

$$23 \cdot 1 - 1 = 22 \text{ is not divisible by 7}$$ $$23 \cdot 4 - 1 = 91 \text{ is divisible by 7}$$

7. Show that $5n - 1$ is divisible by 4

Solution.

$5n - 1$ is not always divisible by 4

For example

$5 \cdot 2 - 1 = 9$ is not divisible by 4

$5 \cdot 1 - 1 = 4$ is divisible by 4

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