diff --git a/chapter_4/exercises.md b/chapter_4/exercises.md index 9799918..3af0e50 100644 --- a/chapter_4/exercises.md +++ b/chapter_4/exercises.md @@ -3011,3 +3011,272 @@ which is a quotient of two integers with a nonzero denominator. Hence it is a rational number. This is what is to be shown. This incorrect prove is assuming what is to be proved. + +--- + +**Exercise Set 4.4** + +Page 220 + +Give a reason for your answer in each of 1-13. Assume that all variables +represent integers. + +1. is $52$ divisible by $13$? + +2. Does $7 \mid 56$? + +3. Does $5 \mid 0$? + +4. Does $3$ divide $(3k + 1)(3k + 2)(3k + 3)$? + +5. Is $6m(2m + 10)$ divisible by $4$? + +6. Is $29$ a multiple of $3$? + +7. Is $-3$ a factor of $66$? + +8. Is $6a(a + b)$ a multiple of $3a$? + +9. Is $4$ a factor of $2a \cdot 34b$? + +10. Does $7 \mid 34$? + +11. Does $13 \mid 73$? + +12. If $n = 4k + 1$, does $8$ divide $n^2 - 1$? + +13. If $n = 4k + 3$, does $8$ divide $n^2 - 1$? + +14. Fill in the blanks in the following proof that for all integers $a$ and $b$, + if $a \mid b$ then $a \mid (-b)$. + +**Proof:** + +Suppose $a$ and $b$ are integers such that __ (a) __. By definition of +divisibility, there exists an integer $r$ such that __ (b) __. By substitution, + +$$ -b = -(ar) = a(-r) $$ + +Let $t = $ __ (c) __. Then $t$ is an integer because $t = (-1) \cdot r$, and +both $-1$ and $r$ are integers. Thus, by substitution, $-b = at$, where $t$ is +an integer, and by the definition of divisibility, __ (d) __, as was to be +shown. + +Prove statements 15 and 16 directly from the definition of divisibility. + +15. For all integers $a$, $b$, and $c$, if $a \mid b$ and $a \mid c$ then + $a \mid (b + c)$. + +16. For all integers $a$, $b$, and $c$, if $a \mid b$ then $a \mid c$ then + $a \mid (b - c)$. + +17. For all integers $a$, $b$, $c$, and $d$, if $a \mid c$ and $b \mid d$ then + $ab \mid cd$. + +18. Consider the following statement: The negative of any multiple of $3$ is a + multiple of $3$. + +a. Write the statement formally using a quantifier and a variable. + +b. Determine whether the statement is true or false and justify your answer. + +19. Show that the following statement is false: For all integers $a$ and $b$, if + $3 \mid (a + b)$ then $3 \mid (a - b)$. + +For each statement in 20-32, determine whether the statement is true or false. +Prove the statement directly from the definitions if it is true, and give a +counterexample if it is false. + +20. The sum of any three consecutive integers is divisible by $3$. + +21. The product of any two even integers is a multiple of $4$. + +22. A necessary condition for an integer to be divisible by $6$ is that it be + divisible by $2$. + +23. A sufficient condition for an integer to be divisible by $8$ is that it be + divisible by $16$. + +24. For all integers $a$, $b$, and $c$, if $a \mid b$ and $a \mid c$ then + $a \mid (2b - 3c)$. + +25. For all integers $a$, $b$, and $c$, if $a$ is a factor of $c$ and $b$ is a + factor of $c$ then $ab$ is a factor of $c$. + +26. For all integers $a$, $b$, and $c$, if $ab \mid c$ then $a \mid c$ and + $b \mid c$. + +27. For all integers $a$, $b$, and $c$, if $a \mid (b + c)$ then $a \mid b$ or + $a \mid c$. + +28. For all integers $a$, $b$, and $c$, if $a \mid bc$ then $a \mid b$ or + $a \mid c$. + +29. For all integers $a$ and $b$, if $a \mid b$ then $a^2 \mid b^2$. + +30. For all integers $a$ and $n$, if $a \mid n^2$ and $a \leq n$ then + $a \mid n$. + +31. For all integers $a$ and $b$, if $a \mid 10b$ then $a \mid 10$ or + $a \mid b$. + +32. A fast-food chain has a contest in which a card with numbers on it is given + to each customer who makes a purchase. If some of the numbers on the card + add up to $100$, then the customer wins $100. A certain customer receives a + card containing the numbers + +72, 21, 15, 36, 69, 81, 9, 27, 42, and 63. + +Will the customer win $100? Why or why not? + +33. Is it possible to have a combination of nickels, dimes, and quarters that + add up to $4.72? Explain. + +34. Consider a string consisting of _a_'s, _b_'s, and _c_'s where the number of + _b_'s is three times the number of _a_'s and the number of _c_'s is five + times the number of _a_'s. Prove that the length of the string is divisible + by $3$. + +35. Two athletes run a circular track at a steady pace so that the first + completes one round in 8 minutes and the second in 10 minutes. If they both + start from the same spot at 4 P.M., when will be the first time they return + to the start? + +36. It can be shown (see exercises 44-48) that an integer is divisible by 3 if, + and only if, the sum of its digits is divisible by 3; an integer is + divisible by 9 if, and only if, the sum of its digits is divisible by 9; an + integer is divisible by 5 if, and only if, its right-most digit is a 5 or a + 0; and an integer is divisible by 4 if, and only if, the number formed by + its right-most two digits is divisible by 4. Check the following integers + for divisibility by 3, 4, 5, and 9. + +a. 637,425,403,705,125 + +b. 12,858,306,120,312 + +c. 517,924,440,926,512 + +d. 14,328,083,360,232 + +37. Use the unique factorization theorem to write the following integers in + standard factored form. + +a. 1,176 + +b. 5,733 + +c. 3,675 + +38. Let $n = 8,424$. + +a. Write the prime factorization for $n$. + +b. Write the prime factorization for $n^5$. + +c. Is $n^5$ divisible by 20? Explain. + +d. What is the least positive integer $m$ so that $8,424 \cdot m$ is a perfect +square? + +39. Suppose that in standard factored form + $a = p_1^{e_1}p_2^{e_2} \dots p_k^{e^k}$, where $k$ is a positive integer; + $p_1, p_2, \dots, p_k$ are prime numbers; and $e_1, e_2, \dots, e_k$ are + positive integers. + +a. What is the standard factored form for $a^3$? + +b. Find the least positive integer $k$ such that +$2^4 \cdot 3^5 \cdot 7 11^2 \cdot k$ is a perfect cube (that is, it equals an +integer to the third power). Write the resulting product as a perfect cube. + +40. + +a. If $a$ and $b$ are integers and $12a = 25b$, does $12 \mid b$? does +$25 \mid a$? Explain. + +b. If $x$ and $y$ are integers and $10x = 9y$, does $10 \mid y$? does +$9 \mid x$? Explain. + +41. How many zeros are at the end of $45^8 \cdot 88^5$? Explain how you can + answer this question without actually computing the number. (_Hint:_ + $10 = 2 \cdot 5$.) + +42. If $n$ is an integer and $n > 1$, then $n!$ is the product of $n$ and every + other positive integer that is less than $n$. For example, + $5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$. + +a. Write $6!$ in standard factored form. + +b. Write $20!$ in standard factored form. + +c. Without computing the value of $(20!)^2$ determine how many zeros are at the +end of this number when it is written in decimal form. Justify your answer. + +43. At a certain university 2/3 of the mathematics students and 3/5 of the + computer science students have taken a discrete mathematics course. The + number of mathematics students who have taken the course equals the number + of computer science students who have taken the course. If there are at + least 100 mathematics students at the university, what are the least + possible number of mathematics students and the least possible number of + computer science students at the university? + +**Definition:** Given any nonnegative integer $n$, the **decimal +representation** of $n$ is an expression of the form + +$$ d_kd_{k + 1} \dots d_2d_1d_0 $$ + +where $kr is a nonnegative integer, $d_0, d_1, d_2, \dots, d_k$ (called the +**decimal digits** of $n$) are integers from $0$ to $9$ inclusive, $dk \neq 0$ +unless $n = 0$ and $k = 0$, and + +$$ n = d_k \cdot 10^k + d_{k + 1} \cdot 10^{k + 1} + \dots + d_2 \cdot 10^2 + d_1 \cdot 10 + d_0 $$ + +(For example, $2,503 = 2 \cdot 10^3 + 5 \cdot 10^2 + 0 \cdot 10 + 3$.) + +44. Prove that if $n$ is any nonnegative integer whose decimal representation + ends in $0$, then $5 \mid n$. (_Hint:_ If the decimal representation of a + nonnegative integer $n$ ends in $d_0$, then $n = 10m + d_0$ for some integer + $m$.) + +45. Prove that if $n$ is any nonnegative integer whose decimal representation + ends in $5$, then $5 \mid n$. + +46. Prove that if the decimal representation of a nonnegative integer $n$ ends + in $d_1d_0$ and if $4 \mid (10d_1 + d_0)$, then $4 \mid n$. (_Hint:_ If the + decimal representation of a nonnegative integer $n$ ends in $d_1d_0$, then + there is an integer $s$ such that $n = 100s + 10d_1 + d_0$.) + +47. Observe that + +$$ +7,524 = 7 \cdot 1,000 + 5 \cdot 100 + 2 \cdot 10 + 4 \\ +\quad = 7(999 + 1) + 5(99 + 1) + 2(9 + 1) + 4 \\ +\quad = (7 \cdot 99 + 7) + (5 \cdot 99 + 5) + (2 \cdot 9 + 2) + 4 \\ +\quad = (7 \cdot 999 + 5 \cdot 99 2 \cdot 9) + (7 + 5 + 2 + 4) \\ +\quad = (7 \cdot 111 \cdot 9 + 5 \cdot 11 \cdot 9 + 2 \cdot 9) + (7 + 5 + 2 + 4) \\ +\quad = (7 \cdot 111 + 5 \cdot 11 + 2) \cdot 9 + (7 + 5 + 2 + 4) \\ +\quad = (\text{an integer divisible by 9})i + (\text{the sum of the digits of } 7,524) +$$ + +Since the sum of the digits of $7,524$ is divisible by $9$, $7,524$ can be +written as a sum of two integers each of which is divisible by $9$. It follows +from exercise 15 that $7,524$ is divisible by $9$. + +Generalize the argument given in this example to any nonnegative integer $n$. In +other words, prove that for any nonnegative integer $n$, if the sum of the +digits of $n$ is divisible by $9r, then $n$ is divisible by $9$. + +48. Prove that for any nonnegative integer $n$, if the sum of the digits of $n$ + is divisible by $3$, then $n$ is divisible by $3$. + +49. Given a positive integer $n$ written in decimal form, the alternating sum of + the digits of $n$ is obtained by starting with the right-most digit, + subtracting the digit immediately to its left, adding the next digit to the + left, subtracting the next digit, and so forth. For example, the alternating + sum of the digits of 180,928 is $8 - 2 + 9 - 0 + 8 - 1 = 22$. Justify the + fact that for any nonnegative integer $n$, if the alternating sum of the + digits of $n$ is divisible by 11, then $n$ is divisible by 11. + +50. The integer 123,123 has the form _abc,abc_, where _a_, _b_, and _c_ are + integers from $0$ through $9$. Consider all six-digit integers of this form. + Which prime numbers divide every one of these integers? Prove your answer. diff --git a/chapter_4/notes.md b/chapter_4/notes.md index 726a991..21a3db5 100644 --- a/chapter_4/notes.md +++ b/chapter_4/notes.md @@ -175,3 +175,221 @@ Page 210 **Corollary 4.2.3** The double of a rational number is rational. + +--- + +Page 213 + +**Definition** + +If $n$ and $d$ are integers then + +$n$ is **divisible by** $d$ if, and only if, $n$ equals $d$ times some integer +and $d \neq 0$. + +Instead of "$n$ is divisible by $d$," we can say that + +$n$ **is a multiple of** $d$, or + +$d$ **is a factor of** $n$, or + +$d$ **is a divisor of** $n$, or + +$d$ **divides** $n$. + +The notation $d \mid n$ is read "$d$ divides $n$." Symbolically, if $n$ and $d$ +are integers: + +$$ d \mid n \Leftrightarrow \exists \text{ an integer, say } k, \text{ such that } n = dk \text{ and } d \neq 0 $$ + +The notation $d \nmid n$ is read "$d$ does not divide $n$." + +--- + +Page 214 + +**Theorem 4.4.1 A Positive Divisor of a Positive Integer** + +For all integers $a$ and $b$, if $a$ and $b$ are positive and $a$ divides $b$ +then $a \leq b$. + +**Proof:** + +Suppose $a$ and $b$ are positive integers such that $a$ divides $b$. _[We must +show that $a \leq b$.]_ By definition of divisibility, there exists an integer +$k$ so that $b = ak$. By property T25 of Appendix A, $k$ must be positive +because both $a$ and $b$ are positive. It follows that + +$$ 1 \leq k $$ + +because every positive integer is greater than or equal to $1$. Multiplying both +sides by $a$ gives + +$$ a \leq ka = b $$ + +because multiplying both sides of an inequality by a positive number preserves +the inequality by property T20 of Appendix A. Thus $a \leq b$ _[as was to be +shown]._ + +--- + +Page 214 + +**Theorem 4.4.2 Divisors of 1** + +The only divisors of $1$ are $1$ and $-1$. + +**Proof:** + +Since $1 \cdot 1 = 1$ and $(-1)(-1) = 1$, both $1$ and $-1$ are divisors of $1$. +Now suppose $m$ is any integer that divides $1$. Then there exists an integer +$n$ such that $1 = mn$. By Theorem T25 in Appendix A, either both $m$ and $n$ +are positive or both $m$ and $n$ are negative. If both $m$ and $n$ are positive, +then $m$ is a positive integer divisor of $1$. By Theorem 4.4.1, $m \leq 1$, +and, since the only positive integer that is less than or equal to $1$ is $1$ +itself, it follows that $m = 1$. On the other hand, if both $m$ and $n$ are +negative, then by Theorem T12 in Appendix A, $(-m)(-n) = mn = 1$. In this case +$-m$ is a positive integer divisor of $1$, and so, by the same reasoning, +$-m = 1$ and thus $m = -1$. Therefore there are only two possibilities: either +$m = 1$ or $m = -1$. So the only divisors of $1$ are $1$ and $-1$. + +--- + +Page 215 + +For all integers $n$ and $d$, $d \nmid n \Leftrightarrow \frac{n}{d}$ is not an +integer. + +--- + +Page 216 + +**Theorem 4.4.3 Transitivity of Divisibility** + +For all integers $a$, $b$, and $c$, if $a$ divides $b$ and $b$ divides $c$, then +$a$ divides $c$. + +**Proof:** + +Suppose $a$, $b$, and $c$ are any _[particular but arbitrarily chosen]_ integers +such that $a$ divides $b$ and $b$ divides $c$. _[We must show that $a$ divides +$c$.]_ By definition of divisibility, + +$$ b = ar \text{ and } c = bs \quad \text{ for some integers } r \text{ and } s $$ + +By substitution + +$$ c = bs $$ + +$$ \quad = (ar)s $$ + +$$ \quad = a(rs) \quad \text{ by basic algebra} $$ + +Let $k = rs$. Then $k$ is an integer since it is a product of integers, and +therefore + +$$ c = ak \quad \text{ where } k \text{ is an integer} $$ + +Thus $a$ divides $c$ by definition of divisibility. _[This is what was to be +shown.]_ + +--- + +Page 217 + +**Theorem 4.4.4 Divisibility by a Prime** + +Any integer $n > 1$ is divisible by a prime number. + +**Proof:** + +Suppose $n$ is a _[particular but _arbitrarily chosen]_ integer that is greater +than $1$. _[We must show that there is a prime number that divides $n$.]_ If $n$ +is prime, then $n$ is divisible by a prime number (namely itself), and we are +done. If $n$ is not prime, the as discussed in Example 4.1.2b, + +$n = r_0s_0$ where $r_0$ and $s_0$ are integers and $1 < r_0 < n$ and +$1 < s_0 < n$. + +It follows by definition of divisibility that $r_0 \mid n$. + +If $r_0$ is prime, then $r_0$ is a prime number that divides $n$, and we are +done. If $r_0$ is not prime, then + +$r_0 = r_1s_1$ where $r_1$ and $s_1$ are integers and $1 < r_1 < r_0$ and +$1 < s_1 < r_0$. + +It follows by the definition of divisibility that $r_q \mid r_0$. But we already +know that $r_0 \mid n$. Consequently, by transitivity of divisibility, +$r_1 \mid n$. + +If $r_1$ is prime, then $r_1$ is a prime number that divides $n$, and we are +done. If $r_1$ is not prime, then + +$r_1 = r_2s_2$ where $r_2$ and $s_2$ are integers and $1 < r_2 < r_1$ and +$1 < s_2 < r_1$. + +It follows by the definition of divisibility that $r_2 \mid r_1$. But we already +know that $r_1 \mid n$. Consequently, by transitivity of divisibility, +$r_2 \mid n$. + +If $r_2$ is prime, then $r_2$ is a prime number that divides $n$, and we are +done. If $r_2$ is not prime, then we may repeat the previous process by +factoring $r_2$ as $r_3s_3$. + +We may continue in this way, factoring successive factors of $n$ until we find a +prime factor. We must succeed in a finite number of steps because each new +factor is both less than the previous one (which is less than $n$) and greater +than $1$, and there are fewer than $n$ integers strictly between $1$ and $n$. +Thus we obtain a sequence + +$$ r_0, r_1, r_2, \dots, r_k $$ + +where $k \geq 0$, $1 < r_k < r_{k - 1} < \dots < r_2 < r_1 < r_0 < n$, and +$r_i \mid n$ for each $i = 0, 1, 2, \dots, k$. The condition for termination is +that $r_k$ should be prime. Hence $r_k$ is a prime number that divides $n$. +_[This is what we were to show.]_ + +--- + +Page 218 + +**Proposed Divisibility Property:** For all integers $a$ and $b$, if $a \mid b$ +and $b \mid a$ then $a = b$. + +**Counterexample:** Let $a = 2$ and $b = -2$. Then $-2 = (-1) \cdot 2$ and +$2 = (-1) \cdot (-2)$, and thus + +$$ a \mid b \text{ and } b \mid a, \text{ but } a \neq b \text{ because } 2 \neq -2 $$ + +Therefore, the statement is false. + +--- + +Page 219 + +**Theorem 4.4.5 Unique Factorization of Integers Theorem (Fundamental Theorem of +Arithmetic)** + +Given any integer $n > 1$, there exist a positive integer $k$, distinct prime +numbers $p_1, p_2, \dots, p_k$, and positive integers $e_1, e_2, \dots, e_k$ +such that + +$$ n = p_1^{e_1}p_2^{e^2}p_3^{e^3}\dots p_k^{e_k} $$ + +and any other expression for $n$ as a product of prime numbers is identical to +this except, perhaps, for the order in which the factors are written. + +--- + +Page 219 + +**Definition** + +Given any integer $n > 1$, the **standard factored form** of $n$ is an +expression of the form + +$$ n = p_1^{e_1}p_2^{e^2}p_3^{e^3}\dots p_k^{e_k} $$ + +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 $p_1 < p_2 < \dots < p_k$. diff --git a/chapter_4/test_yourself.md b/chapter_4/test_yourself.md index db6f218..2ed658c 100644 --- a/chapter_4/test_yourself.md +++ b/chapter_4/test_yourself.md @@ -102,3 +102,33 @@ real number; not rational zero is an integer that is a ratio of integers where the denominator is not zero, $0 = \dfrac{0}{1}$. + +--- + +**Test Yourself** + +Page 220 + +1. To show that a nonzero integer $d$ divides an integer $n$, we must show that + ______. + +2. To say that $d$ divides $n$ means the same as saying that ______ is divisible + by ______. + +3. If $a$ and $b$ are positive integers and $a \mid b$, then ______ is less than + or equal to ______. + +4. For all integers $n$ and $d$, $d \nmid n$ if, and only if, ______. + +5. If $a$ and $b$ are integers, the notation $a \mid b$ denotes ______ and the + notation $a/b$ denotes ______. + +6. The transitivity of divisibility theorem says that for all integers $a$, $b$, + and $c$, if ______ then ______. + +7. The divisibility by a prime theorem says that every integer greater than $1$ + is ______. + +8. The unique factorization of integers theorem says that any integer greater + than $1$ is either ______ or can be written as ______ in a way that is unique + except possibly for the ______ in which the numbers are written.