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234
chapter_4/exercises.md
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@ -3999,3 +3999,237 @@ Omitted.
Which prime numbers divide every one of these integers? Prove your answer.
Omitted.
---
**Exercise Set 4.5**
Page 232
For each of the values of $n$ and $d$ given in 1-6, find integers $q$ and $r$
such that $n = dq + r$ and $0 \leq r < d$.
1. $n = 70$, $d = 9$
2. $n = 62$, $d = 7$
3. $n =36$, $d = 40$
4. $n = 3$, $d = 11$
5. $n = -45$, $d = 11$
6. $n = -27$, $d = 8$
**Evaluate the expressions in 7-10.**
7.
a. $43\ div\ 9$
b. $43 \mod 9$
8.
a. $50\ div\ 7$
b. $50 \mod 7$
9.
a. $28\ div\ 5$
b. $28 \mod 5$
10.
a. $30\ div\ 2$
b. $30 \mod 2$
11. Check the correctness of formula (4.5.1) given in Example 4.5.3 for the
following values of $\text{Day}T$ and $N$.
a. $\text{Day}T = 6(\text{Saturday}) \text{ and } N = 15$
b. $\text{Day}T = 0(\text{Sunday}) \text{ and } N = 7$
c. $\text{Day}T = 4(\text{Thursday}) \text{ and } N = 12$
12. Justify formula (4.5.1) for general values of $\text{Day}T$ and $N$.
13. On a Monday a friend says he will meet you again in 30 days. What day of the
week will that be?
14. If today is Tuesday, what day of the week will it be 1,000 days from today?
15. January 1, 2000, was a Saturday, and 2000 was a leap year. What day of the
week will January 1, 2050, be?
16. Suppose $d$ is a positive and $n$ is any integer. If $d \mid n$, what is the
remainder obtained when the quotient remainder theorem is applied to $n$
with divisor $d$?
17. Prove directly from the definitions that for every integer $n$,
$n^2 - n + 3$ is odd. Use division into two cases: $n$ is even and $n$ is
odd.
18.
a. Prove that the product of any two consecutive integers is even.
b. The result of part (a) suggests that the second approach in the discussion of
Example 4.5.7 might be possible after all. Write a new proof of Theorem 4.5.3
based on this observation.
19. Prove directly from the definitions that for all integers $m$ and $n$, if
$m$ and $n$ have the same parity, then $5m + 7n$ is even. Divide into two
cases: $m$ and $n$ are both even and $m$ and $n$ are both odd.
20. Suppose $a$ is any integer. If $a \mod 7 = 4$, what is $5a \mod 7$? In other
words, if division of $a$ by $7$ gives a remainder of $4$, what is the
remainder when $5a$ is divided by $7$? Your solution should show that you
obtain the same answer no matter what integer you start with.
21. Suppose $b$ is any integer. If $b \mod 12 = 5$, what is $8b \mod 12$? In
other words, if division of $b$ by $12$ gives a remainder of $5$, what is
the remainder when $8b$ is divided by $12$? Your solution should show that
you obtain the same answer no matter what integer you start with.
22. Suppose $c$ is any integer. If $c \mod 15 = 3$, what is $10c \mod 15$? In
other words, if division of $c$ by $15$ gives a remainder of $3$, what is
the remainder when $10c$ is divided by $15$? Your solution should show that
you obtain the same answer no matter what integer you start with.
23. Prove that for every integer $n$, if $n \mod 5 = 3$ then $n^2 \mod 5 = 4$.
24. Prove that for all integers $m$ and $n$, if $m \mod 5 = 2$ and
$n \mod 5 = 1$ then $mn \mod 5 = 2$.
25. Prove that for all integers $a$ and $b$, if $a \mod 7 = 5$ and
$b \mod 7 = 6$ then $ab \mod 7 = 2$.
26. Prove that a necessary and sufficient condition for an integer $n$ to be
divisible by a positive integer $d$ is that $n \mod d = 0$.
27. Use the quotient-remainder theorem with divisor equal to $2$ to prove that
the square of any integer can be written in one of the two forms $4k$ or
$4k + 1$ for some integer $k$.
28.
a. Prove: Given any set 9f three consecutive integers, one of the integers is a
multiple of $3$.
b. Use the result of part (a) to prove that any product of three consecutive
integers is a multiple of 3.
29.
a. Use the quotient-remainder theorem with divisor equal to $3$ to prove that
the square of any integer has the form $3k$ or $3k + 1$ for some integer $k$.
b. Use the $\mod$ notation to rewrite the result of part (a).
30.
a. Use the quotient-remainder theorem with divisor equal to $3$ to prove that
the product of any two consecutive integers has the form $3k$ or $3k + 2$ for
some integer $k$.
b. Use the $\mod$ notation to rewrite the result of part (a).
In 32-33, you may use the properties listed in Example 4.3.3.
31.
a. Prove that for all integers $m$ and $n$, $m + n$ and $m - n$ are either both
odd or both even.
b. Find all solutions to the equation $m^2 - n^2 = 56$ for which both $m$ and
$n$ are positive integers.
c. Find all solutions to the equation $m^2 - n^2 = 88$ for which both $m$ and
$n$ are positive integers.
32. Given any integers $a$, $b$, and $c$, if $a - b$ is even and $b - c$ is
even, what can you say about the parity of $2a - (b + c)$? Prove your
answer.
33. Given any integers $a$, $b$, and $c$, if $a - b$ is 9dd and $b - c$ is even,
what can you say about the parity of $a - c$? Prove your answer.
34. Given any integer $n$, if $n > 3$, could $n$, $n + 2$, and $n + 4$ all be
prime? Prove or give a counterexample.
Prove each of the statements in 35-43.
35. The fourth power of any integer has the form $8m$ or $8m + 1$ for some
integer $m$.
36. The product of any four consecutive integers is divisible by $8$.
37. For any integer $n$, $n^2 + 5$ is not divisible by $4$.
38. For every integer $m$, $m^2 = 5k$, or $m^w = 5k + 1$, or $m^2 = 5k + 4$ for
some integer $k$.
39. Every prime number except $2$ and $3$ has the form $6q + 1$ or $6q + 5$ for
some integer $q$.
40. If $n$ is any odd integer, then $n^4 \mod 16 = 1$.
41. For all real numbers $x$ and $y$, $|x| \cdot |y| = |xy|$.
42. For all real numbers $r$ and $c$ with $c \geq 0$, $-c \leq r \leq c$ if, and
only if, $|r| \leq c$. _(Hint: Proving $A$ if, and only if, $B$ requires
proving both if $A$ then $B$ and if $B$ then $A$.)_
43. For all real numbers $a$ and $b$, $\lvert|a| - |b|\rvert \leq |a - b|$.
44. A matrix $\mathbb{M}$ has 3 rows and 4 columns.
$$
\left[\begin{array}{cccc}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{21} & a_{22} & a_{23} & a_{24} \\
a_{31} & a_{32} & a_{33} & a_{34} \\
\end{array}\right]
$$
The 12 entries in the matrix are to be stored in _row major_ form in locations
7,609 to 7,620 in a computer's memory. This means that the entries in the first
row (reading left to right) are stored first, then the entries in the second
row, and finally the entries in the third row.
a. Which location will $a_{22}$ be stored in?
b. Write a formula (in $i$ and $j$) that gives the integer $n$ so that $a_{ij}$
is stored in location 7,609 + $n$.
c. Find formulas (in $n$) for $r$ and $s$ so that $a_{rs}$ is stored in location
7.609 + n.
45. Let $\mathbb{M}$ be a matrix with $m$ rows and $n$ columns, and suppose that
the entries of $\mathbb{M}$ are stored in a computer's memory in row major
form (see exercise 44) in locations $N$, $N + 1$, $N + 2$, $\dots$,
$N + mn - 1$. Find formulas in $k$ for $r$ and $s$ so that $a_{rs}$ is
stored in location $N + k$.
46. If $m$, $n$, and $d$ are integers, $d > 0$ and $m \mod d = n \mod d$, does
it necessarily follow that $m = n$? That $m - n$ is divisible by $d$? Prove
your answers.
47. If $m$, $n$, and $d$ are integers $d > 0$, and $d \mid (m - n)$, what is
the relation between $m 'mod d'$ and $n \mod d$? Prove your answer.
48. If $m$, $n$, $a$, $b$, and $d$ are integers, $d > 0$ and $m \mod d = a$ and
$n \mod d = b$, is $(m + n) \mod d = a + b$? Is
$(m + n) \mod d = (a + b) \mod d$? Prove your answers.
49. If $m$, $n$, $a$, $b$, and $d$ are integers, $d > 0$, and $m \mod d = a$ and
$n \mod d = b$, is $(mn) \mod d = ab$? Is $(mn) \mod d = ab \mod d$? Prove
your answers.
50. Prove that if $m$, $d$, and $k$ are integers and $d > 0$, then
$(m + dk) \mod d = m \mod d$.

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chapter_4/notes.md
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@ -393,3 +393,268 @@ $$ n = p_1^{e_1}p_2^{e^2}p_3^{e^3}\dots p_k^{e_k} $$
where $n$ is a positive integer, $p_1,p_2,\dots , p_k$ are prime numbers,
$e_1,e_2,\dots ,e_k$ are positive integers, and $p_1 < p_2 < \dots < p_k$.
---
Page 223
**Theorem 4.5.1 The Quotient Remainder Theorem**
Given any integer $n$ and positive integer $d$, there exists unique integers $q$
and $r$ such that
$$ n = dq + r \quad \text{ and } \quad 0 \leq r < d $$
---
Page 224
**Definition**
Given an integer $n$ and a positive integer $d$,
$$ n\ div\ d = \text{ the integer quotient obtained when } n \text{ is divided by } d \text{ and } $$
$$ n \mod d = \text{ the nonnegative integer remainder obtained when } n \text{ is divided by } d $$
Symbolically, if $n$ and $d$ are integers and $d > 0$ then
$$ n\ div\ d = \quad \text{ and } \quad n \mod d = r \Leftrightarrow n = dq + r $$
where $q$ and $r$ are integers and $0 \leq r < d$.
---
**Theorem 4.5.2 The Parity Property**
Any two consecutive integers have opposite parity.
**Proof:**
Suppose that two _[particular but arbitrarily chosen]_ consecutive integers are
given; call them $m$ and $m + 1$. _[We must show that one of $m$ and $m + 1$ is
even and that the other is odd.]_ By the parity property, either $m$ is even or
$m$ is odd. _[We break the proof into two cases depending on whether $m$ is even
or odd.]_
_Case 1 ($m$ is even):_ In this case, $m = 2k$ for some integer $k$, and so
$m + 1 = 2k + 1$, which is odd _[by the definition of odd.]_ Hence in this case,
one of $m$ and $m + 1$ is even and the other is odd.
_Case 2 ($m$ is odd):_ In this case, $m = 2k + 1$ for some integer $k$, and so
$m + 1 = (2k + 1) + 1 = 2k + 2 = 2(k + 1)$. But $k + 1$ is an integer because it
is a sum of two integers. Therefore, $m + 1$ equals twice some integer, and thus
$m + 1$ is even. Hence in this case also, one of $m$ and $m + 1$ is even and the
other is odd.
It follows that regardless of which case actually occurs for the particular $m$
and $m + 1$ is even and the other is odd. _[This is what was to be shown.]_
---
Page 227
**Method of Proof by Division into Cases**
To prove a statement of the form "If $A_1$ or $A_2$ or $\dots$ or $A_n$, then
$C$," prove all of the following:
$$
\text{If } A_1, \text{ then } C \\
\text{If } A_2, \text{ then } C \\
\vdots \\
\text{If } A_n, \text{ then } C \\
$$
This process shows that $C$ is true regardless of which of $A_1$, $A_2$,
$\dots$, $A_n$ happens to be the case.
---
Page 229
**Theorem 4.5.3**
The square of any odd integer has the form $8m + 1$ for some integer $m$.
**Proof:** Suppose $n$ is a _[particular but arbitrarily chosen]_ odd integer.
By the quotient-remainder theorem with the divisor equal to $4$, $n$ can be
written in one of the forms
$$ 4q \quad \text{ or } \quad 4q + 1 \quad \text{ or } \quad 4q + 2 \quad \text{ or } \quad 4q + 3 $$
for some integer $q$. In fact, since $n$ is odd and $4q$ and $4q + 2$ are even,
$n$ must have one of the forms
$$ 4q + 1 \quad \text{ or } \quad 4q + 3 $$
_Case 1($n = 4q + 1$ for some integer $q$)_ _[We must find an integer $m$ such
that $n^2 = 8m + 1$.]_ Since $n = 4q + 1$,
$$ n^2 = (4q + 1)^2 \quad \text{ by substitution} $$
$$ \quad = (4q + 1)(4q + 1) \quad \text{ by definition of square} $$
$$ \quad = 16q^2 + 8q + 1 $$
$$ \quad = 8(2q^2 + 1) + 1 \quad \text{ by the laws of algebra} $$
Let $m = 2q^2 + q$. Then $m$ is an integer since $2$ and $q$ are integers and
sums and products of integers are integers. Thus, substituting,
$$ n^2 = 8m + 1 \quad \text{ where } m \text{ is an integer} $$
_Case 2 ($n = 4q + 3$ for some integer $q$):_ _[We must find an integer $m$ such
that $n^2 = 8m + 1$.]_ Since $n = 4q + 3$,
$$ n^2 = (4q + 3)^2 \quad \text{ by substitution} $$
$$ \quad = (4q + 3)(4q + 3) \quad \text{ by definition of square} $$
$$ \quad = 16q^2 + 24q + 9 $$
$$ \quad = 16q^2 + 24q + (8 + 1) $$
$$ \quad = 8(2q^2 + 3q + 1) + 1 \quad \text{ by the laws of algebra} $$
_[The motivation for the choice of algebra steps was the desire to write the
expression in the form $8 \cdot \text{ some integer } + 1$.]_
Let $m = 2q^2 + 3q + 1$. Then $m$ is an integer since $1$, $2$, $3$, and $q$ are
integers and sums and products of integers are integers. Thus, substituting,
$$ n^2 = 8m + 1 \quad \text{ where } m \text{ is an integer} $$
Cases 1 and 2 show that given any odd integer, whether of the form $4q + 1$ or
$4q + 3$, $n^2 = 8m + 1$ for some integer $m$. _[This is what we needed to
show.]_
---
Page 231
**Definition**
For any real number $x$, the **absolute value of** $x$, denoted $|x|$, is
defined as follows:
$$
|x| =
\begin{cases}
x & \text{if } x \geq 0 \\
-x & \text{if } x < 0
\end{cases}
$$
---
Page 231
**Lemma 4.5.4**
For every real number, $r$, $-|r| \leq r \leq |r|$
**Proof:** Suppose $r$ is any real number. We divide into cases according to
whether $r = 0$, $r > 0$, or $r < 0$.
_Case 1($r = 0$):_ In this case, by definition of absolute value, $|r| = r = 0$
since $0 = -0$, we have that $-0 = -|r| = 0 = r = |r|$, and so it is true that
$$ -|r| \leq r \leq |r| $$
_Case 2 ($r > 0$):_ In this case, by definition of absolute value,
[$\&|\text{pipe}|r|\text{pipe}||=|r\&$]. Also, since $r$ is positive and $-|r|$
is negative, $-|r| < r$. Thus it is true that
$$ -|r| \leq r \leq |r| $$
_Case 3 ($r < 0$):_ In this case, by definition of absolute value, $|r| = -r$.
Multiplying both sides by $-1$ gives that $-|r| = r$. Also, since $r$ is
negative and $|r|$ is positive, $r < |r|$. Thus it is also true in this case
that
$$ -|r| \leq r \leq |r| $$
Hence, in every case,
$$ -|r| \leq r \leq |r| $$
_[as was to be shown]._
---
Page 231
**Lemma 4.5.5**
For ever real number $r$, $|-r| = |r|$.
**Proof:** Suppose $r$ is any real number. By Theorem T23 in Appendix A, if
$r > 0$, then $-r < 0$, and if $r < 0$, then $-r > 0$. Thus
$$
|-r| =
\begin{cases}
-r & \text{if } -r > 0 \\
0 & \text{if } -r = 0 \\
-(-r) & \text{if } -r < 0
\end{cases}
$$
$$
\quad =
\begin{cases}
-r & \text{if } -r > 0 \\
0 & \text{if } r = 0 \\
r & \text{if } -r < 0
\end{cases}
$$
$$
\quad =
\begin{cases}
-r & \text{if } r < 0 \\
0 & \text{if } r = 0 \\
r & \text{if } r > 0
\end{cases}
$$
$$
\quad =
\begin{cases}
r & \text{if } r \geq 0 \\
-r & \text{if } r < 0 \\
\end{cases}
$$
$$ \quad = |r| $$
---
Page 231
**Theorem 4.5.6 The Triangle Inequality**
For all real numbers $x$ and $y$, $|x + y| \leq |x| + |y|$.
**Proof:** Suppose $x$ and $y$ are real numbers.
_Case 1 ($x + y \geq 0$):_ In this case, $|x + y| = x + y$, and so, by Lemma
4.5.4,
$$ x \leq |x| \quad \text{ and } y \leq |y| $$
Hence, by Theorem T26 of Appendix A,
$$ |x + y| = x + y \leq |x| + |y| $$
_Case 2 ($x + y < 0$):_ In this case, $|x + y| = -(x + y) = (-x) + (-y)$, and
so, by Lemmas 4.5.4 and 4.5.5,
$$ -x \leq |-x| = |x| \quad \text{ and } \quad -y \leq |-y| = |y| $$
It follows, by Theorem T26 of Appendix A, that
$$ |x + y| = (-x) + (-y) \leq |x| + |y| $$
Hence in both cases $|x + y| = |x| + |y|$ _[as was to be shown]._

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chapter_4/test_yourself.md
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@ -148,3 +148,26 @@ divisible by some prime number.
except possibly for the ______ in which the numbers are written.
prime; a product of prime numbers; order
---
**Test Yourself**
Page 232
1. The quotient-remainder theorem says that for all integers $n$ and $d$ with
$d \geq 0$, there exists ______ $q$ and $r$ such that ______ and ______.
2. If $n$ and $d$ are integers with $d > 0$, $n\ div\ d$ is ______ and
$n \mod d$ is ______.
3. The parity of an integer indicates whether the integer is ______.
4. According to the quotient-remainder theorem, if an integer $n$ is divided by
a positive integer $d$, the possible remainders are ______. This implies that
$n$ can be written in one of the forms ______ for some integer $q$.
5. To prove a statement of the form "If $A_1$ or $A_2$ or $A_3$, then $C$,"
prove ______ and ______ and ______.
6. The triangle inequality says that for all real numbers $x$ and $y$, ______.