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🚧 In mid of 4.10

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132
chapter_4/exercises.md
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@ -9175,10 +9175,14 @@ executed.
$i := 2\\ \text{\textbf{if }} (i > 3 \text{ or } i \leq 0)\\ \ \ \ \ \text{\textbf{then }} z := 1\\ \ \ \ \ \text{\textbf{else }} z := 0$
$z = 0$
2.
$i := 3\\ \text{\textbf{if }} (i \leq 3 \text{ or } i > 6)\\ \ \ \ \ \text{\textbf{then }} z := 2\\ \ \ \ \ \text{\textbf{else }} z := 0$
$z = 2$
3. Consider the following algorithm segment:
$\text{\textbf{if }} x \cdot y > 0 \text{\textbf{ then do }} y := 3 \cdot x\\ \ \ \ \ x := x + 1 \text{\textbf{end do}}\\ \ \ \ \ z := x \cdot y$
@ -9188,8 +9192,12 @@ below.
a. $x = 2, y = 3$
$y = 3 \cdot 3 = 9$, and $x = 2 + 1 = 3$, and $z = 9 \cdot 3 \cdot = 27$
b. $x = 1, y = 1$
$y = 3 \cdot 1 = 3$, and $x = 1 + 1 = 2$, and $z = 3 \cdot 2 = 6$
Find the values of $a$ and $e$ after execution of the loops in 4 and 5 by first
making trace tables for them.
@ -9197,17 +9205,50 @@ making trace tables for them.
$a := 2\\ \text{\textbf{for }} i := 1 \text{\textbf{ to }} 3\\ \ \ \ \ a:= 3a + 1\\ \text{\textbf{next }} i$
| | 0 | 1 | 2 | 3 |
| --- | - | - | -- | -- |
| $a$ | 2 | 7 | 22 | 67 |
| $i$ | 1 | 2 | 3 | 4 |
After execution, $a = 67$.
5.
$e := 2, f := 0\\ \text{\textbf{for }} k := 1 \text{\textbf{ to }} 3\\ \ \ \ \ e := e \cdot k\\ \ \ \ \ f := e + f\\ \text{\textbf{next }} k$
| | 0 | 1 | 2 | 3 |
| --- | - | - | - | -- |
| $e$ | 2 | 2 | 4 | 12 |
| $f$ | 0 | 2 | 6 | 18 |
| $k$ | 1 | 2 | 3 | 4 |
After execution, $e = 12$, $f = 18$.
Make a trace table to trace the action of Algorithm 4.10.1 for the input
variables given in 6 and 7.
6. $a = 26, d = 7$
| | 0 | 1 | 2 | 3 |
| --- | -- | -- | -- | -- |
| $a$ | 26 | 26 | 26 | 26 |
| $d$ | 7 | 7 | 7 | 7 |
| $r$ | 26 | 19 | 12 | 5 |
| $q$ | 0 | 1 | 2 | 3 |
After execution, $q = 3$, and $r = 5$.
7. $a = 59, d = 13$
| | 0 | 1 | 2 | 3 | 4 |
| --- | -- | -- | -- | -- | -- |
| $a$ | 59 | 59 | 59 | 59 | 59 |
| $d$ | 13 | 13 | 13 | 13 | 13 |
| $r$ | 59 | 46 | 33 | 20 | 7 |
| $q$ | 0 | 1 | 2 | 3 | 4 |
After execution, $q = 4$, $r = 7$.
8. The following algorithm segment makes change; given an amount of money $A$
between 1¢ and 99¢, it determines a breakdown of $A$ into quarters $(q)$,
dimes $(d)$, nickels $(n)$, and pennies $(p)$.
@ -9223,39 +9264,104 @@ $$
a. Trace this algorithm segment for $A = 69$.
| | | | | |
| --- | -- | -- | - | - |
| $A$ | 69 | 19 | 9 | |
| $q$ | 2 | | | |
| $d$ | | 1 | | |
| $n$ | | | 1 | |
| $p$ | | | | 4 |
b. Trace this algorithm segment for $A = 87$.
| | | | | |
| --- | -- | -- | - | - |
| $A$ | 87 | 12 | 2 | |
| $q$ | 3 | | | |
| $d$ | | 1 | | |
| $n$ | | | 0 | |
| $p$ | | | | 0 |
Find the greatest common divisor of each of the pairs of integers in 9-12. (Use
any method you wish.)
9. $27$ and $72$
$$ \text{gcd}(27, 72) = 9 $$
10. $5$ and $9$
$$ \text{gcd}(5, 9) = 1 $$
11. $7$ and $21$
$$ \text{gcd}(7, 21) = 7 $$
12. $48$ and $54$
$$ \text{gcd}(54, 48) = \text{gcd}(48, 6) = \text{gcd}(6, 0) = 6 $$
Use the Euclidean algorithm to hand-calculate the greatest common divisors of
each of the pairs of itnegers in 13-16.
13. $1,188$ and $385$
$$ \text{gcd}(1188, 385) = \text{gcd}(385, 33) = \text{gcd}(33, 22) = \text{gcd}(22, 11) = \text{gcd}(11, 0) = 11 $$
14. $509$ and $1,177$
$$ \text{gcd}(1177, 509) = \text{gcd}(509, 159) = \text{gcd}(159, 32) = \text{gcd}(32, 31) = \text{gcd}(31, 1) = \text{gcd}(1, 0) = 1 $$
15. $832$ and $10,933$
$$ \text{gcd}(10933, 832) = \text{gcd}(832, 117) = \text{gcd}(117, 13) = \text{gcd}(13, 0) = 13 $$
16. $4,131$ and $2,431$
$$ \text{gcd}(4131, 2431) = \text{gcd}(2431, 1700) = \text{gcd}(1700, 731) = \text{gcd}(731, 238) = \text{gcd}(238, 17) = \text{gcd}(17, 0) = 17 $$
Make a trace table to trace the action of Algorithm 4.10.2 for the input
variables given in 17-19.
17. $1,001$ and $871$
| | | | | | | | |
| ------------ | ---- | --- | --- | -- | -- | -- | -- |
| $A$ | 1001 | | | | | | |
| $B$ | 871 | | | | | | |
| $a$ | 1001 | 871 | 130 | 91 | 39 | 13 | |
| $b$ | 871 | 130 | 91 | 39 | 13 | 0 | |
| $r$ | 871 | 130 | 91 | 39 | 13 | 0 | |
| $\text{gcd}$ | | | | | | | 13 |
After execution, $\text{gcd} = 13$.
18. $5,859$ and $1,232$
| | | | | | | | | |
| ------------ | ---- | ---- | --- | --- | -- | -- | - | - |
| $A$ | 5859 | | | | | | | |
| $B$ | 1232 | | | | | | | |
| $a$ | 5859 | 1232 | 931 | 301 | 28 | 21 | 7 | |
| $b$ | 1232 | 931 | 301 | 28 | 21 | 7 | 0 | |
| $r$ | 1232 | 931 | 301 | 28 | 21 | 7 | 0 | |
| $\text{gcd}$ | | | | | | | | 7 |
After execution, $\text{gcd} = 7$.
19. $1,570$ and $488$
| | | | | | | | | | |
| ------------ | ---- | --- | --- | -- | -- | -- | -- | - | - |
| $A$ | 1570 | | | | | | | | |
| $B$ | 488 | | | | | | | | |
| $a$ | 1570 | 488 | 106 | 64 | 42 | 22 | 20 | 2 | |
| $b$ | 488 | 106 | 64 | 42 | 22 | 20 | 2 | 0 | |
| $r$ | 488 | 106 | 64 | 42 | 22 | 20 | 2 | 0 | |
| $\text{gcd}$ | | | | | | | | | 2 |
After execution, $\text{gcd} = 2$.
**Definition: Integers $a$ and $b$ are said to be **relatively prime** if, and
only if, their greatest common divisor is $1$.
@ -9264,8 +9370,34 @@ integers are relatively prime.
20. $4,167$ and $2,563$
| | | | | | | | | | | |
| ------------ | ---- | ---- | ---- | --- | --- | --- | -- | - | - | - |
| $A$ | 4167 | | | | | | | | | |
| $B$ | 2563 | | | | | | | | | |
| $a$ | 4167 | 2563 | 1604 | 959 | 645 | 314 | 17 | 8 | 1 | |
| $b$ | 2563 | 1604 | 959 | 645 | 314 | 17 | 8 | 1 | 0 | |
| $r$ | 2563 | 1604 | 959 | 645 | 314 | 17 | 8 | 1 | 0 | |
| $\text{gcd}$ | | | | | | | | | | 1 |
After execution, $\text{gcd} = 1$, and because of this, $4,167$ and $2,563$ are
_relatively_ prime.
21. $34,391$ and $6,728$
| | | | | | | | | | |
| ------------ | ----- | ---- | --- | --- | -- | - | - | - | - |
| $A$ | 34391 | | | | | | | | |
| $B$ | 6728 | | | | | | | | |
| $a$ | 34391 | 6728 | 751 | 720 | 31 | 7 | 3 | 1 | |
| $b$ | 6728 | 751 | 720 | 31 | 7 | 3 | 1 | 0 | |
| $r$ | 6728 | 751 | 720 | 31 | 7 | 3 | 1 | 0 | |
| $\text{gcd}$ | | | | | | | | | 1 |
After execution, $\text{gcd} = 1$, and because of this, $34,391$ and $6,728$ are
_relatively_ prime.
RESUME HERE
22. Prove that for all positive integers $a$ and $b$, $a \mid b$ if, and only
if, $\text{gcd}(a, b) = a$. (Note that to prove "$A$ if, and only if, $B$,"
you need to prove "if $A$ then $B$" and "if $B$ then $A$.")

31
chapter_4/test_yourself.md
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@ -294,10 +294,16 @@ Page 277
1. When an algorithm statement of the form $x := e$ is executed, ______.
The expression $e$ is evaluated (using the current values of all the variables
in the expression), and this value is placed in the memory location
corresponding to $x$ (replacing any previous contents of the location)
2. Consider an algorithm statement of the following form.
$\text{\textbf{if }(condition)}\\ \text{\textbf{then }} s_1\\ \text{\textbf{else }} s_2$
then the algorithm at $s_1$ is executed; then the algorithm at $s_2$ is executed
When such a statement is executed, the truth or falsity of the _condition_ is
evaluated. If _condition_ is true, ______. If _condition_ is false, ______.
@ -312,8 +318,18 @@ $\text{\textbf{end while}}$
When such a statement is executed, the truth or falsity of the _condition_ is
evaluated. If _condition_ is true, ______. If _condition_ is false, ______.
the statements that make up the body of the loop are executed in order and then
execution moves back to the beginning of the loop and the process repeats; the
loop ends and execution passes to the next algorithm statement following the
loop
4. Consider an algorithm statement of the following form.
the statements that make up the body of the loop are executed in order, the
variable's value is assigned to the next (iterated), and then execution moves
back to the beginning of the loop and the process repeats; the loop ends and
execution passes to the next algorithm statement following the loop
$\text{\textbf{for } variable } := \text{initial expression \textbf{to} final expression.}$
_[statements that make up the body of the loop]_
@ -328,13 +344,24 @@ ______. If not, ______.
5. Given a nonnegative integer $a$ and a positive integer $d$ the division
algorithm computes ______.
Integers $q$ and $r$ with the property that $n = dq + r$ and $0 \leq r < d$.
6. Given integers $a$ and $b$, not both zero, $\text{gcd}(a, b)$ is the integer
$d$ that satisfies the following two conditions: ______ and ______.
$d \mid a$ and $d \mid b$; if $c$ is a common divisor of both $a$ and $b$, then
$c \leq d$.
7. If $r$ is a positive integer, then $gcd(r, 0) =$ ______.
$r$
8. If $a$ and $b$ are integers not both zero and if $q$ and $r$ are nonnegative
integers such that $a = bq + r$ then $\text{gcd}(a,b ) =$ ______.
integers such that $a = bq + r$ then $\text{gcd}(a, b) =$ ______.
$gcd(b, r)$
9. Given positive integers $A$ and $B$ with $A > B$, the Euclidean algorithm
computes.
computes ______.
the greatest common divisor of $A$ and $B$, $\text{gcd}(A, B)$.