diff --git a/chapter_4/exercises.md b/chapter_4/exercises.md index 606a931..f0cb650 100644 --- a/chapter_4/exercises.md +++ b/chapter_4/exercises.md @@ -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 + $(m + dk) \mod d = m \mod d$. diff --git a/chapter_4/notes.md b/chapter_4/notes.md index 21a3db5..4a61016 100644 --- a/chapter_4/notes.md +++ b/chapter_4/notes.md @@ -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]._ diff --git a/chapter_4/test_yourself.md b/chapter_4/test_yourself.md index e03d261..d546509 100644 --- a/chapter_4/test_yourself.md +++ b/chapter_4/test_yourself.md @@ -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 $y$, ______.