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