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🚧 Setup for chapter 4.10

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chapter_4/exercises.md
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@ -9161,3 +9161,236 @@ See Page 266.
degrees? Explain. degrees? Explain.
Omitted. Omitted.
---
**Exercise Set 4.10**
Page 278
Find the value of $z$ when each of the algorithm segments in 1 and 2 is
executed.
1.
$i := 2\\ \text{\textbf{if }} (i > 3 \text{ or } i \leq 0)\\ \ \ \ \ \text{\textbf{then }} z := 1\\ \ \ \ \ \text{\textbf{else }} z := 0$
2.
$i := 3\\ \text{\textbf{if }} (i \leq 3 \text{ or } i > 6)\\ \ \ \ \ \text{\textbf{then }} z := 2\\ \ \ \ \ \text{\textbf{else }} z := 0$
3. Consider the following algorithm segment:
$\text{\textbf{if }} x \cdot y > 0 \text{\textbf{ then do }} y := 3 \cdot x\\ \ \ \ \ x := x + 1 \text{\textbf{end do}}\\ \ \ \ \ z := x \cdot y$
Find the value of $z$ if prior to execution $x$ and $y$ have the values given
below.
a. $x = 2, y = 3$
b. $x = 1, y = 1$
Find the values of $a$ and $e$ after execution of the loops in 4 and 5 by first
making trace tables for them.
4.
$a := 2\\ \text{\textbf{for }} i := 1 \text{\textbf{ to }} 3\\ \ \ \ \ a:= 3a + 1\\ \text{\textbf{next }} i$
5.
$e := 2, f := 0\\ \text{\textbf{for }} k := 1 \text{\textbf{ to }} 3\\ \ \ \ \ e := e \cdot k\\ \ \ \ \ f := e + f\\ \text{\textbf{next }} k$
Make a trace table to trace the action of Algorithm 4.10.1 for the input
variables given in 6 and 7.
6. $a = 26, d = 7$
7. $a = 59, d = 13$
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 $(p)$.
$$
q := A \text{div } 25 \\
A := A \mod 25 \\
d := A \text{div } 10 \\
A := A \mod 10 \\
n := A \text{div } 5 \\
p := A \mod 5
$$
a. Trace this algorithm segment for $A = 69$.
b. Trace this algorithm segment for $A = 87$.
Find the greatest common divisor of each of the pairs of integers in 9-12. (Use
any method you wish.)
9. $27$ and $72$
10. $5$ and $9$
11. $7$ and $21$
12. $48$ and $54$
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 $385$
14. $509$ and $1,177$
15. $832$ and $10,933$
16. $4,131$ and $2,431$
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 $871$
18. $5,859$ and $1,232$
19. $1,570$ and $488$
**Definition: Integers $a$ and $b$ are said to be **relatively prime** if, and
only if, their greatest common divisor is $1$.
In 20 and 21 trace the action of Algorithm 4.10.2 to determine whether the
integers are relatively prime.
20. $4,167$ and $2,563$
21. $34,391$ and $6,728$
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 $A$.")
23.
a. Prove that if $a$ and $b$ are integers, not both zero, and
$d = \text{gcd}(a, b)$, then $\dfrac{a}{d}$ and $\dfrac{b}{d}$ are integers with
no common divisor that is greater than $1$.
b. Write an algorithm that accepts the numerator and denominator of a fraction
as input and produces as output the numerator and denominator of that fraction
written in lowest terms. (The algorithm may call upon the Euclidean algorithm as
needed.)
24. Complete the proof of Lemma 4.10.2 by proving the following: If $a$ and $b$
are any integers with $b \neq 0$ and $q$ and $r$ are any integers such that
$$ a = bq + r $$
then
$$ \text{gcd}(b, r) \leq \text{gcd}(a, b) $$
25.
a. Prove: If $a$ and $d$ are positive integers and $q$ and $r$ are integers such
that $a = dq + r$ and $0 < r < d$, then
$$ -a = d(-(q + 1)) + (d - r) $$
and
$$ 0 < d - r < d $$
b. Indicate how to modify Algorithm 4.10.1 to allow for the input $a$ to be
negative.
26.
a. Prove that if $a$, $d$, $q$, and $r$ are integers such that $a = dq + r$ and
$0 \leq r < d$, then
$$ q = \left\lfloor \frac{a}{d} \right\rfloor \quad \text{ and } r = a - \left\lfloor \frac{a}{d} \right\rfloor \cdot d$$
b. In a computer language with a built-in floor function, $\text{div}$ and
$\mod$ can be calculated as follows:
$$ a \text{div } d = \left\lfoor \frac{a}{d} \right\rfloor \quad \text{ and } \quad a \mod d = a - \left\lfloor \frac{a}{d} \right\rfloor \cdot d $$
Rewrite the steps of Algorithm 4.10.2 for a computer language with a built-in
floor function but without $\text{div}$ and $\mod$.
27. An alternative to the Euclidean algorithm uses subtraction rather than
division to compute greatest common divisors. (After all, division is
repeated subtraction.) It is based on the following lemma.
**Lemma 4.10.3**
**Algorithm 4.10.3 Computing gcd's by Subtraction**
_[Given two positive integers $A$ and $B$, variables $a$ and $b$ are set equal
to $A$ and $B$. Then a repetitive process begins. If $a \neq 0$, and $b \neq 0$,
then the larger of $a$ and $b$ is set equal to
$a - b (\text{if } a \geq b) \text{ or to } b - a(\text{if } a < b)$, and the
smaller of $a$ and $b$ is left unchanged. This process is repeated over and over
until eventually $a$ or $b$ becomes $0$. By Lemma 4.10.3, after each repetition
of the process,_
$$ \text{gcd}(A, B) = \text{gcd}(a, b) $$
_After the last repetition,_
$$ \text{gcd}(A, B) = \text{gcd}(a, 0) \quad \text{ or } \quad \text{gcd}(A, B) = \text{gcd}(0, b) $$
_depending on whether $a$ or $b$ is nonzero. But by Lemma 4.10.1,_
$$ \text{gcd}(a, 0) = a \quad \text{ and } \quad \text{gcd}(0, b) = b $$
_Hence, after the last repetition,_
$$ \text{gcd}(A, B) = a \text{ if } a \neq 0 \quad \text{ or } \quad \text{gcd}(A, B) = b \text{ if } b \neq 0 $$
**Input:** $A, B$ _[positive integers]_
**Algorithm Body:**
$a := A, b := B\\ \text{\textbf{while }} (a \neq 0 \text{ and } b \neq 0)\\ \ \ \ \ \text{\textbf{if }} a \geq b \text{\textbf{ then }} a := a - b\\ \ \ \ \ \ \ \ \ \text{\textbf{else }} b := b - a\\ \text{\textbf{end while}}\\ \ \ \ \ \text{\textbf{if }} a = 0 \text{\textbf{ then }} gcd := b\\ \ \ \ \ \text{\textbf{else }} gcd := a$
_[After execution of the **if-then-else** statement,
$\text{gcd} = \text{gcd}(A, B)$.]_
**Output:** $\text{gcd}$ _[a positive integer]_
a. Prove Lemma 4.10.3.
b. Trace the execution of Algorithm 4.10.3 for $A = 360$ and $B = 336$.
c. Trace the execution of Algorithm 4.10.3 for $A = 768$ and $B = 348$.
Exercises 28-32 refer to the following definition.
**Definition:** The **least common multiple** of two nonzero integers $a$ and
$b$, denoted $\text{\textbf{lcm}}(a, b)$, is the positive integer $c$ such that
a. $a \mid c$ and $b \mid c$
b. for all positive integers $m$, if $a \mid m$ and $b \mid m$, then $c \leq m$.
28. Find
a. $\text{lcm}(12, 18)$
b. $\text{lcm}(2^2 \cdot 3 \cdot 5, 2^3 \cdot 3^2)$
c. $\text{lcm}(2800, 6125)$
29. Prove that for all positive integers $a$ and $b$,
$\text{gcd}(a, b) = \text{lcm}(a, b)$ if, and only if, $a = b$.
30. Prove that for all positive integers $a$ and $b$, $a \mid b$ if, and only
if, $\text{lcm}(a, b) = b$.
31. Prove that for all integers $a$ and $b$,
$\text{gcd}(a, b) \mid \text{lcm}(a, b)$.
32. Prove that for all positive integers $a$ and $b$,
$\text{gcd}(a, b) \cdot \text{lcm}(a, b) = ab$.

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@ -1242,3 +1242,252 @@ such a way that
3. there is no edge from any one vertex of $V$ to any other vertex of $V$; 3. there is no edge from any one vertex of $V$ to any other vertex of $V$;
4. there is no edge from any one vertex of $W$ to any other vertex of $W$. 4. there is no edge from any one vertex of $W$ to any other vertex of $W$.
---
Page 268
Execution of an **if-then-else** statement occurs as follows:
1. The _condition_ is evaluated by substituting the current values of all
algorithm variables appearing in it and evaluating the truth or falsity of
the resulting statement.
2. If _condition_ is true, then $s_1$ is executed and execution moves to the
next algorithm statement following the **if-then-else** statement.
3. If _condition_ is false, then $s_2$ is executed and execution moves to the
next algorithm statement following the **if-then-else** statement.
---
Page 269
Execution of a **while** loop occurs as follows:
1. The _condition_ is evaluated by substituting the current values of all the
algorithm variables and evaluating the truth or falsity of the resulting
statement.
2. If _condition_ is true, all the statements in the body of the loop are
executed in order. Then execution moves back to the beginning of the loop and
the process repeats.
3. If _condition_ is false, execution passes to the next algorithm statement
following the loop.
---
Page 270
A **for-next** loop is executed as follows:
1. The **for-next** loop _variable_ is set equal to the value of _initial
expression_.
2. A check is made to determine whether the value of _variable_ is less than or
equal to the value of _final expression_.
3. If the value of _variable_ is less than or equal to the value of _final
expression_, then the statements in the body of the loop are executed in
order, _variable_ is increased by $1$, and execution returns back to step 2.
4. If the value of _variable_ is greater than the value of _final expression_,
then execution passes to the next algorithm statement following the loop.
---
Page 272
**Algorithm 4.10.1 Division Algorithm**
_[Given a nonnegative integer $a$ and a positive integer $d$, the aim of the
algorithm is to find integers $q$ and $r$ that satisfy the conditions
$a = dq + r$ and $0 \leq r < d$. This is done by subtracting $d$ repeatedly from
$a$ until the result is less than $d$ but is still nonnegative._
$$ 0 \leq a - d - d - d - \dots - d = a - dq < d$$
_The total number of $d$'s that are subtracted is the quotient $q$. The quantity
$a - dq$ equals the remainder $r$.]_
**Input:** _$a$ [a nonnegative integer], $d$ [a positive integer]_
**Algorithm Body:**
$$ r:= a, q := 0 $$
_[Repeatedly subtract $d$ from $r$ until a number less than $d$ is obtained. Add
$1$ to $q$ each time $d$ is subtracted.]_
$\text{\textbf{while}} (r \geq d)\\ \ \ \ \ r := r- d\\ \ \ \ \ q:= q + 1\\ \text{\textbf{end while}}$
_[After execution of the $\text{\textbf{while}}$ loop, $a = dq + r$.]_
**Output:** $q$, $r$ _[nonnegative integers]_
---
Page 273
**Definition**
Let $a$ and $b$ be integers that are not both zero. The **greatest common
divisor** of $a$ and $b$, denoted $\text{\textbf{gcd}}(a, b)$, is that integer
$d$ with the following properties:
1. $d$ is a common divisor of both $a$ and $b$. In other words,
$$ d \mid a \quad \text{ and } \quad d \mid b $$
2. For every integer $c$, if $c$ is a common divisor of both $a$ and $b$, then
$c$ is less than or equal to $d$. In other words,
$$ \text{for every integer } c, \text{ if } c \mid a \text{ and } c \mid b \text{ then } c \leq d$$
---
Page 274
**Lemma 4.10.1**
If $r$ is a positive integer, then $\text{gcd}(r, 0) = r$.
**Proof:** Suppose $r$ is a positive integer. _[We must show that the greatest
common divisor of both $r$ and $0$ is $r$.]_ Certainly, $r$ is a common divisor
of both $r$ and $0$ because $r$ divides itself and also $r$ divides $0$ (since
every positive integer divides $0$). Also no integer larger than $r$ can be a
common divisor of $r$ and $0$ (since no integer larger than $r$ can divide $r$).
Hence $r$ is the greatest common divisor of $r$ and $0$.
---
Page 274
**Lemma 4.10.2**
If $a$ and $b$ are any integers not both zero, and if $q$ and $r$ are any
integers such that
$$ a = bq + r $$
then
$$ \text{gcd}(a, b) = \text{gcd}(b, r) $$
**Proof:** _[The proof is divided into two sections: (1) proof that
$\text{gcd}(a, b) \leq \text{gcd}(b, r)$, and (2) proof that
$\text{gcd}(b, r) \leq \text{gcd}(a, b)$. Since each $\text{gcd}$ is less than
or equal to the other, the two must be equal.]_
1. $\text{\textbf{gcd}}(a, b) \leq \text{\textbf{gcd}}(b, r)$:
a. _[We will first show that any common divisor of $a$ and $b$ is also a common
divisor of $b$ and $r$.]_
Let $a$ and $b$ be integers, not both zero, and let $c$ be a common divisor of
$a$ and $b$. Then $c \mid a$ and $c \mid b$, and so, by definition of
divisibility, $a = nc$ and $b = mc$, for some integers $n$ and $m$. Substitute
into the equation
$$ a = bq + r $$
to obtain
$$ nc = (mc)q + r $$
Then solve for $r$:
$$ r = nc - (mc)q = (n - mq)c $$
Now $n - mq$ is an integer, and so, by definition of divisibility, $c \mid r$.
Because we already know that $c \mid b$, we can conclude that $c$ is a common
divisor of $b$ and $r$ _[as was to be shown]_.
b. _[Next we show that $\text{gcd}(a, b) \leq \text{gcd}(b, r)$.]_
Now the greatest common divisor of $a$ and $b$ defined because $a$ and $b$ are
not both zero. Also, by part (a), every common divisor of $a$ and $b$ is a
common divisor of $b$ and $r$, and so the greatest common divisor of $a$ and $b$
is a common divisor of $b$ and $r$. But then $\text{gcd}(a, b)$ (being one of
the common divisors of $b$ and $r$) is less than or equal to the greatest common
divisor of $b$ and $r$:
$$ \text{gcd}(a, b) \leq \text{gcd}(b, r) $$
2. $\text{\textbf{gcd}}(b, r) \leq \text{\textbf{gcd}}(a, b)$:
The second part of the proof is very similar to the first part. It is left as an
exercise.
---
Page 275
**Euclidean Algorithm Description**
1. Let $A$ and $B$ be integers with $A > B \geq 0$.
2. To find the greatest common divisor of $A$ and $B$, first check whether
$B = 0$. If it is, then $\text{gcd}(A, B) = A$ by Lemma 4.10.1. If it isn't,
then $B > 0$ and the quotient-remainder theorem can be used to divide $A$ by
$B$ to obtain a quotient $q$ and a remainder $r$:
$$ A = Bq + r \quad \text{ where } \quad 0 \leq r < B $$
By Lemma 4.10.2, $\text{gcd}(A, B) = \text{gcd}(B, r)$. Thus the problem of
finding the greatest common divisor of $A$ and $B$ is reduced to the problem of
finding the greatest common divisor of $B$ and $r$.
_[What makes this information useful is the fact that the larger number of the
pair $(B, r)$ is smaller than the larger number of the pair $(A, B)$. The reason
is that the value of $r$ found by the quotient-remainder theorem satisfies_
$$ 0 \leq r < B $$
_And, since by assumption $B < A$, we have that_
$$ 0 \leq r < B < A $$
_]_.
3. Now just repeat the process, starting again at (2), but use $B$ instead of
$A$ and $r$ instead of $B$. The repetitions are guaranteed to terminate
eventually with $r = 0$ because each new remainder is less than the preceding
one and all are nonnegative.
---
Page 277
**Algorithm 4.10.2 Euclidean Algorithm**
_[Given two integers $A$ and $B$ with $A > B \geq 0$, this algorithm computes
$\text{gcd}(A, B)$. It is based on two facts:_
1. $\text{gcd}(a, b) = \text{gcd}(b, r)$ _if $a$, $b$, $q$, and $r$ are integers
with $b = b \cdot q + r$ and $0 \leq r < b$._
2. $\text{gcd}(a, 0) = a$._]_
**Input:** $A, B$ _[integers with $A > B \geq 0$]_
**Algorithm Body:**
$a := A, b := B, r:= B$
_[If $b \neq 0$, compute $a \mod b$, the remainder of the integer division of
$a$ by $b$, and set $r$ equal to this value. Then repeat the process using $b$
in place of $a$ and $r$ in place of $b$.]_
$\text{\textbf{while }} (b \neq 0)\\ \ \ \ \ r:= a \mod b$
_[The value of $a \mod b$ can be obtained by calling the division algorithm.]_
$\ \ \ \ a := b\\ \ \ \ \ b := r\\ \text{\textbf{end while}}$
_[After execution of the $\text{\textbf{while}}$ loop, $\text{gcd}(A, B) = a$.]_
$\text{gcd} := a$
**Output:** $\text{gcd}$ _[a positive integer]_

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chapter_4/test_yourself.md
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@ -285,3 +285,56 @@ for each pair of vertices
any one vertex of $W$ to any other vertex of $W$. any one vertex of $W$ to any other vertex of $W$.
one edge; no edge; no edge one edge; no edge; no edge
---
**Test Yourself**
Page 277
1. When an algorithm statement of the form $x := e$ is executed, ______.
2. Consider an algorithm statement of the following form.
$\text{\textbf{if }(condition)}\\ \text{\textbf{then }} s_1\\ \text{\textbf{else }} s_2$
When such a statement is executed, the truth or falsity of the _condition_ is
evaluated. If _condition_ is true, ______. If _condition_ is false, ______.
3. Consider an algorithm statement of the following form.
$\text{\textbf{while }(condition)}$
_[statements that make up the body of the loop]_
$\text{\textbf{end while}}$
When such a statement is executed, the truth or falsity of the _condition_ is
evaluated. If _condition_ is true, ______. If _condition_ is false, ______.
4. Consider an algorithm statement of the following form.
$\text{\textbf{for } variable } := \text{initial expression \textbf{to} final expression.}$
_[statements that make up the body of the loop]_
$\text{\textbf{next } (same) variable}$
When such a statement is executed, _variable_ is set equal to the value of the
_initial expression_, and a check is made to determine whether the value of
_variable_ is less than or equal to the value of _final expression_. If so,
______. If not, ______.
5. Given a nonnegative integer $a$ and a positive integer $d$ the division
algorithm computes ______.
6. Given integers $a$ and $b$, not both zero, $\text{gcd}(a, b)$ is the integer
$d$ that satisfies the following two conditions: ______ and ______.
7. If $r$ is a positive integer, then $gcd(r, 0) =$ ______.
8. If $a$ and $b$ are integers not both zero and if $q$ and $r$ are nonnegative
integers such that $a = bq + r$ then $\text{gcd}(a,b ) =$ ______.
9. Given positive integers $A$ and $B$ with $A > B$, the Euclidean algorithm
computes.