DISCRETE MATHEMATICS-CONNECT ACCESS ONLY
8th Edition
ISBN: 9781264309696
Author: ROSEN
Publisher: MCG
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Textbook Question
Chapter 4.3, Problem 7E
Express in pseudocode the trial division algorithm for determining whether an integer is prime.
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Construct a know-show table for each statement below that appears to be true.
Problem 3. Pricing a multi-stock option the Margrabe formula
The purpose of this problem is to price a swap option in a 2-stock model, similarly as
what we did in the example in the lectures. We consider a two-dimensional Brownian
motion given by W₁ = (W(¹), W(2)) on a probability space (Q, F,P). Two stock prices
are modeled by the following equations:
dX
=
dY₁ =
X₁ (rdt+
rdt+0₁dW!)
(²)),
Y₁ (rdt+dW+0zdW!"),
with Xo
xo and Yo =yo. This corresponds to the multi-stock model studied in class,
but with notation (X+, Y₁) instead of (S(1), S(2)). Given the model above, the measure
P is already the risk-neutral measure (Both stocks have rate of return r). We write
σ = 0₁+0%. We consider a swap option, which gives you the right, at time T, to
exchange one share of X for one share of Y. That is, the option has payoff
F=(Yr-XT).
(a) We first assume that r = 0 (for questions (a)-(f)). Write an explicit expression for
the process Xt.
Reminder before proceeding to question (b): Girsanov's theorem…
Problem 1. Multi-stock model
We consider a 2-stock model similar to the one studied in class. Namely, we consider
=
S(1)
S(2)
=
S(¹) exp (σ1B(1) + (M1 - 0/1 )
S(²) exp (02B(2) + (H₂-
M2
where (B(¹) ) +20 and (B(2) ) +≥o are two Brownian motions, with
t≥0
Cov (B(¹), B(2)) = p min{t, s}.
"
The purpose of this problem is to prove that there indeed exists a 2-dimensional Brownian
motion (W+)+20 (W(1), W(2))+20 such that
=
S(1)
S(2)
=
=
S(¹) exp (011W(¹) + (μ₁ - 01/1) t)
롱)
S(²) exp (021W (1) + 022W(2) + (112 - 03/01/12) t).
where σ11, 21, 22 are constants to be determined (as functions of σ1, σ2, p).
Hint: The constants will follow the formulas developed in the lectures.
(a) To show existence of (Ŵ+), first write the expression for both W. (¹) and W (2)
functions of (B(1), B(²)).
as
(b) Using the formulas obtained in (a), show that the process (WA) is actually a 2-
dimensional standard Brownian motion (i.e. show that each component is normal,
with mean 0, variance t, and that their…
Chapter 4 Solutions
DISCRETE MATHEMATICS-CONNECT ACCESS ONLY
Ch. 4.1 - Prob. 1ECh. 4.1 - Prove that if a is an integer other than o, then 1...Ch. 4.1 - Theorem 1 Let a, b, and c be integers, where ao ....Ch. 4.1 - Prove that part (iii) of Theorem 1 is true. Let...Ch. 4.1 - Show that if a|b|a, where a and b are integers,...Ch. 4.1 - Show that if a, b, c, and d are integers, where a...Ch. 4.1 - Show that if a, b, and c are integers, where ao ,...Ch. 4.1 - Prove or disprove that if a|bc, where a,b, and c...Ch. 4.1 - Prove that if a and b are integers and a divides...Ch. 4.1 - Prove that if a and b are nonzero integers, a...
Ch. 4.1 - Prove that if a is and integer that is not...Ch. 4.1 - Prove that if a is positive integer, then 4 does...Ch. 4.1 - What are the quotient and remainder when a) 19 is...Ch. 4.1 - What are the quotient and remainder when 44 is...Ch. 4.1 - What time does a 12-hour clock read a) 80 hours...Ch. 4.1 - What time does a 24-hour clock read a) 100 hours...Ch. 4.1 - Suppose that a and b are integers, a4(mod13) , and...Ch. 4.1 - Suppose that a and b are integers, a11(mod19) and...Ch. 4.1 - Show that if a and d are positive integers, then...Ch. 4.1 - Prove or disprove that if a, b, and d are integers...Ch. 4.1 - Let m be a positive integer. Show that a=b(modm)...Ch. 4.1 - Let m be a positive integer. Show that amodm=bmodm...Ch. 4.1 - Show that if n and k are positive integers, then...Ch. 4.1 - Show that if a is and integer d is and integer...Ch. 4.1 - Find a formula of the integer with smallest...Ch. 4.1 - Evaluate these quantities. -17 mod 2 144 mod 7...Ch. 4.1 - Evaluate these quantities. 13 mod 3 -97 mod 11 155...Ch. 4.1 - Find a div m and a mod m when a=111,m=99 ....Ch. 4.1 - Find a div m and a mod m when a=228,m=119 ....Ch. 4.1 - Find the integer a such that a43(mod23) and 22a0 ....Ch. 4.1 - Find the integer a such that a15(mod27) and 26a0 ....Ch. 4.1 - List five integers that are congruent to 4 modulo...Ch. 4.1 - List all integers between -100 and 100 that are...Ch. 4.1 - Decide whether each of these integers is congruent...Ch. 4.1 - Decide whether each of these integers is congruent...Ch. 4.1 - Find each of these values....Ch. 4.1 - Find each of these values. a)...Ch. 4.1 - Find each of these values. a) (192mod41)mod9 b) (...Ch. 4.1 - Find each of these values. a) ( 992mod32)3mod15 b)...Ch. 4.1 - Show that if a = b (mod m) and c= d (mod m), where...Ch. 4.1 - Prob. 41ECh. 4.1 - Show that if a, b, c, and m are integers such that...Ch. 4.1 - Find counter Examples to each of these statements...Ch. 4.1 - Show that if n is an integer then n20 or 1 (mod...Ch. 4.1 - Prob. 45ECh. 4.1 - Prove that if n is and odd positive integer, then...Ch. 4.1 - Prob. 47ECh. 4.1 - Show that Zmwith addition modulo m, where m2 is an...Ch. 4.1 - Prob. 49ECh. 4.1 - Show that the distributive property of...Ch. 4.1 - Write out the addition and multiplication tables...Ch. 4.1 - Write out the addition and multiplication tables...Ch. 4.1 - Determine whether each of the functions f(a)=adivd...Ch. 4.2 - Convert the decimal expansion of each of these...Ch. 4.2 - Convert the decimal expansion of each of these...Ch. 4.2 - Convert the binary expansion of each of these...Ch. 4.2 - Convert the binary expansion of each of these...Ch. 4.2 - Convert the octal expansion of each of these...Ch. 4.2 - Convert the binary expansion of each of these...Ch. 4.2 - Convert the hexadecimal expansion of each of these...Ch. 4.2 - Convert (BADFACED)16 from its hexadecimal...Ch. 4.2 - Convert (ABCDEF)16 from its hexadecimal expansion...Ch. 4.2 - Convert each of the integers in Exercise 6 from a...Ch. 4.2 - Convert (101101111011)2from its binary expansion...Ch. 4.2 - Convert (1 1000 0110 0011)2from its binary...Ch. 4.2 - Show that the hexadecimal expansion of a positive...Ch. 4.2 - Show that the binary expansion of a positive...Ch. 4.2 - Show that the octal expansion of a positive...Ch. 4.2 - Show that the binary expansion of a positive...Ch. 4.2 - Convert (7345321)8 to its binary expansion and (10...Ch. 4.2 - Give a procedure for converting from the...Ch. 4.2 - Give a procedure for converting from the octal...Ch. 4.2 - Explain how to convert from binary to base 64...Ch. 4.2 - Find the sum and the product of each of these...Ch. 4.2 - Find the sum and the product of each of these...Ch. 4.2 - Find the sum and the product of each of these...Ch. 4.2 - Find the sum and the product of each of these...Ch. 4.2 - Use Algorithm 5 to find 7644mod 645.Ch. 4.2 - Use Algorithm 5 to find 11644 mod 645.Ch. 4.2 - Use Algorithm 5 to find 32003mod 99.Ch. 4.2 - Use Algorithm 5 to find 1231001mod 101.Ch. 4.2 - Prob. 29ECh. 4.2 - It Can be shown that every integer can be uniquely...Ch. 4.2 - Show that a positive integer is divisible by 3 if...Ch. 4.2 - Show that a positive integer is divisible by 11 if...Ch. 4.2 - Show that a positive integer is divisible by 3 if...Ch. 4.2 - Prob. 34ECh. 4.2 - Prob. 35ECh. 4.2 - Prob. 36ECh. 4.2 - Prob. 37ECh. 4.2 - Find the decimal expansion of the number with the...Ch. 4.2 - Find the decimal expansion of the number the 3n...Ch. 4.2 - Find the one's complement representations, using...Ch. 4.2 - What integer does each of the following one's...Ch. 4.2 - Prob. 42ECh. 4.2 - Prob. 43ECh. 4.2 - Prob. 44ECh. 4.2 - Prob. 45ECh. 4.2 - Answer Exercise 40, but this time find the two’s...Ch. 4.2 - Answer Exercise 41, if each expansion is tow’s...Ch. 4.2 - Prob. 48ECh. 4.2 - Prob. 49ECh. 4.2 - Answer Exercise 44 for two’s complement...Ch. 4.2 - Show that the integer m with two’s complement...Ch. 4.2 - Prob. 52ECh. 4.2 - Prob. 53ECh. 4.2 - Find the Cantor expansions of a) 2. b) 7. c) 19....Ch. 4.2 - Prob. 55ECh. 4.2 - Describe an algorithm to add two integers from...Ch. 4.2 - Prob. 57ECh. 4.2 - Multiply (1110)2 and (1010)2 by working through...Ch. 4.2 - Describe an algorithm for finding the difference...Ch. 4.2 - Estimate the number of bit operations used to...Ch. 4.2 - Devise an algorithm that, given the binary...Ch. 4.2 - Prob. 62ECh. 4.2 - Prob. 63ECh. 4.2 - Prob. 64ECh. 4.2 - Show that algorithm 4 uses O (q log a) bit...Ch. 4.3 - Determine whether each of these integers is prime....Ch. 4.3 - Determine whether each of these integers is prime....Ch. 4.3 - Find the prime factorization of each of these...Ch. 4.3 - Find the prime factorization of each of these...Ch. 4.3 - Find the prime factorization of 10!.Ch. 4.3 - How may zeros are there at the end of 100!.Ch. 4.3 - Express in pseudocode the trial division algorithm...Ch. 4.3 - Express in pseudocode the algorithm described in...Ch. 4.3 - Show that am+1 is composite if a and m are...Ch. 4.3 - Show that if 2m+1 is and odd prime, then m=2n for...Ch. 4.3 - Show that log2 3 is an irrational number. Recall...Ch. 4.3 - Prove that for every positive integer n, there are...Ch. 4.3 - Prove or disprove that here are three consecutive...Ch. 4.3 - Which positive integers less than 12 are...Ch. 4.3 - Which positive integers less than 30 are...Ch. 4.3 - Determine whether the integers in each of these...Ch. 4.3 - Determine whether the integers in each of these...Ch. 4.3 - We call a positive integer perfect if it equals...Ch. 4.3 - Show that if 2n-1 is prime, then n is prime....Ch. 4.3 - Prob. 20ECh. 4.3 - Find these values of the Euler -function. a) (4)...Ch. 4.3 - Show that n is prime if and only if (n)=n1 .Ch. 4.3 - What is the value of (pk) when p is prime and k is...Ch. 4.3 - What are the greatest common divisors of these...Ch. 4.3 - What are the greatest common divisors of these...Ch. 4.3 - What is the least common multiple of each pair in...Ch. 4.3 - What is the least common multiple of each pair in...Ch. 4.3 - Find god (1000, 625) and 1cm (1000, 625) and...Ch. 4.3 - Find gcd(92928, 123552) and lcm (92928, 123552),...Ch. 4.3 - If the product of tow integers is 273852711 and...Ch. 4.3 - Show that if a and b are positive integers, then...Ch. 4.3 - Use the Euclidean algorithm to find a) gcd(l, 5)....Ch. 4.3 - Use the Euclidean algorithm to find gcd(12, 18)....Ch. 4.3 - How many divisions are required to find gcd(21,...Ch. 4.3 - How many divisions are required to find gcd(34,...Ch. 4.3 - Show that if a and b are positive integers, then...Ch. 4.3 - Show that if a and b are positive integers, then...Ch. 4.3 - Use Exercise 37 to show that the integers...Ch. 4.3 - Using the method followed in Example17, express...Ch. 4.3 - Using the method followed in Example 17, express...Ch. 4.3 - Use the extended Euclidean algorithm to express...Ch. 4.3 - Use the extended Euclidean algorithm to express...Ch. 4.3 - Use the extended Euclidean algorithm to express...Ch. 4.3 - Use the extended Euclidean algorithm to express...Ch. 4.3 - Prob. 45ECh. 4.3 - Find the smallest positive integer with exactly n...Ch. 4.3 - Can you find a formula or rule for the nth terms...Ch. 4.3 - Can you find a formula or rule for the nth term of...Ch. 4.3 - Prove that the product of any three consecutive...Ch. 4.3 - Show that if a, b, and m are integers such that m...Ch. 4.3 - Prove or disprove that n2-79n+1601 is prime...Ch. 4.3 - Prob. 52ECh. 4.3 - Show that there is a composite integer in every...Ch. 4.3 - Adapt the proof in the text that there are...Ch. 4.3 - Adapt the proof in the text that there are...Ch. 4.3 - Prob. 56ECh. 4.3 - Prob. 57ECh. 4.4 - Show that 15 is an inverse of 7 modulo 26Ch. 4.4 - Show that 937 is an inverse of 13 modulo 2436.Ch. 4.4 - By inspection (as discussed prior to Example 1),...Ch. 4.4 - By inspection (as discussed prior to Example 1),...Ch. 4.4 - Find an inverse of a modulo m for each of these...Ch. 4.4 - Find an inverse of a modulo m for each of these...Ch. 4.4 - Show that if a and m are relatively prime positive...Ch. 4.4 - Show that an inverse of a modulo m, where a is an...Ch. 4.4 - Solve the congruence 4x5(mod9) using the inverse...Ch. 4.4 - Solve the congruence 2x7(mod17) using the inverse...Ch. 4.4 - Solve each of these congruences using the modular...Ch. 4.4 - Solve each of these congruences using the modular...Ch. 4.4 - Find the solutions of the congruence...Ch. 4.4 - Find the solutions of the congruence...Ch. 4.4 - Show that if m is an integer greater than 1 and...Ch. 4.4 - a) Show that the positive integers less than 11,...Ch. 4.4 - Show that if p is prime, the only solutions of...Ch. 4.4 - a) Generalize the result in part (a) of Exercise...Ch. 4.4 - This exercise outline a proof of Fermat’s little...Ch. 4.4 - Use the construction the proof of the Chinese...Ch. 4.4 - Use the construction the proof of the Chinese...Ch. 4.4 - Solve the system of congruence x3(mod6) and...Ch. 4.4 - Solve the system of congruence in Exercise 20...Ch. 4.4 - Prob. 24ECh. 4.4 - Write out in pseudocode an algorithm for solving a...Ch. 4.4 - Find all solutions, if any, to the system of...Ch. 4.4 - Find all solutions, if any, to the system of...Ch. 4.4 - Use the Chinese remainder theorem to show that an...Ch. 4.4 - Prob. 29ECh. 4.4 - Complete the proof of the Chinese remainder...Ch. 4.4 - Which integers leave a remainder of 1 when divided...Ch. 4.4 - Which integers divisible by 5 but leave a...Ch. 4.4 - Use Fermat’s little theorem to find 7121 mod 13.Ch. 4.4 - Use Fermat’s little theorem to find 231002 mod 41.Ch. 4.4 - Use Fermat’s little theorem to show that if p is...Ch. 4.4 - Use Exercise 35 to find an inverse of 5 modulo 41.Ch. 4.4 - a) Show that 23401(mod11) by Fermat's little...Ch. 4.4 - a) Use Fermat’s little theorem to compute 3302 mod...Ch. 4.4 - a) Use Fermat’s little theorem to compute 52003...Ch. 4.4 - Show with the help of Fermat’s little theorem that...Ch. 4.4 - Show that if p is and odd prime, then every...Ch. 4.4 - Use Exercise 41 to determine whether M13=213-18191...Ch. 4.4 - Use Exercise 41 to deter mine whether...Ch. 4.4 - Show that if n is prime and b is a positive...Ch. 4.4 - Show that 2047 is a strong pseudoprime to the base...Ch. 4.4 - Show that 1729 is a Carmichael number.Ch. 4.4 - Show that 2821 is a Carmichael number.Ch. 4.4 - Show that if n=p1p2pk , where p1,p2,,pk are...Ch. 4.4 - Use Exercise 48 to show that every integer of the...Ch. 4.4 - Find the nonnegative integer a less than 28...Ch. 4.4 - Express each nonnegative integer a less than 15 a...Ch. 4.4 - Explain how to use the pairs found in Exercise 51...Ch. 4.4 - Solve the system of congruences that arises in...Ch. 4.4 - Show that 2 is a primitive root of 19.Ch. 4.4 - Find the discrete logarithms of 5 and 6 to the...Ch. 4.4 - Let p be and odd prime and r a primitive root of...Ch. 4.4 - Write out a table of discrete logarithms modulo 17...Ch. 4.4 - Which integers are quadratic residues of 11?Ch. 4.4 - Show that if p is an odd prime and a is an integer...Ch. 4.4 - Show that if p is and odd prime, then there are...Ch. 4.4 - Show that if p is and odd prime and a and b are...Ch. 4.4 - Prove Euler’s criterion, which states that if p is...Ch. 4.4 - Prob. 63ECh. 4.4 - Show that if p is an odd prime, then -1 is a...Ch. 4.4 - Find all solutions of the congruence x2=29(mod35)...Ch. 4.4 - Prob. 66ECh. 4.4 - Prob. 67ECh. 4.5 - Which memory locations are assigned by the hashing...Ch. 4.5 - Which memory locations are assigned by the hashing...Ch. 4.5 - A parking lot has 31 visitor space, numbered from...Ch. 4.5 - Use the double hashing procedure we have described...Ch. 4.5 - Prob. 5ECh. 4.5 - Prob. 6ECh. 4.5 - Prob. 7ECh. 4.5 - Write an algorithm in pseudocode for generating a...Ch. 4.5 - Find the first eight terms of the sequence of...Ch. 4.5 - Explain why both 3792 and 2916 would be bad...Ch. 4.5 - Find the sequence of pseudorandom numbers...Ch. 4.5 - Find the sequence of pseudorandom number generated...Ch. 4.5 - Suppose you received these bit strings over a...Ch. 4.5 - Prove that a parity check bit can detect an error...Ch. 4.5 - The first nine digits of the ISBN-10 of the...Ch. 4.5 - The ISBN-10 of the sixth edition of Elementary...Ch. 4.5 - Determine whether the check digit of the ISBN-10...Ch. 4.5 - Find the check digit for the USPS money orders...Ch. 4.5 - Determine whether each of these numbers is a valid...Ch. 4.5 - One digit in each of these identification numbers...Ch. 4.5 - One digit in each of these identification numbers...Ch. 4.5 - Determine which single digit errors are detected...Ch. 4.5 - Determine which transposition errors are detected...Ch. 4.5 - Determine the check digit for the UPCs that have...Ch. 4.5 - Determine whether each of the strings of 12 digits...Ch. 4.5 - Does the check digit of a UPC code detect all...Ch. 4.5 - Determine which transposition errors the check...Ch. 4.5 - Find the check digit a15 that follows each of...Ch. 4.5 - Determine whether each of these 15-digit numbers...Ch. 4.5 - Which errors in a single digit of a 15-digit...Ch. 4.5 - Can the accident transposition of two consecutive...Ch. 4.5 - For each of these initial seven digits of an ISSN,...Ch. 4.5 - Are each of these eight-digit codes possible...Ch. 4.5 - Does the check digit of an ISSN detect every...Ch. 4.5 - Does the check digit of an ISSN detect every error...Ch. 4.6 - Encrypt the message DO NOT PASS GO by translating...Ch. 4.6 - Encrypt the message STOP POLLUTION by translating...Ch. 4.6 - Encrypt the message WATCH YOUR SETEP by...Ch. 4.6 - Decrypt these messages that were encrypted using...Ch. 4.6 - Decrypt these messages encrypted using the shift...Ch. 4.6 - Suppose that when a long string of text is...Ch. 4.6 - Suppose that when a string of English text is...Ch. 4.6 - Suppose that the ciphertext DVE CFMV KF NFEUVI,...Ch. 4.6 - Suppose that the ciphertext ERC WY JJMGMIRXPC...Ch. 4.6 - Determine whether there is a key of which the...Ch. 4.6 - What is the decryption function for an affine...Ch. 4.6 - Find all pairs of integers key (a, b) for affine...Ch. 4.6 - Suppose that the most common letter and the second...Ch. 4.6 - Prob. 14ECh. 4.6 - Decrypt the message EABW EFRO ATMAR ASIN, which is...Ch. 4.6 - Prob. 16ECh. 4.6 - Suppose you have intercepted a ciphertext message...Ch. 4.6 - Use the Vigenère cipher with key BLUE to encrypt...Ch. 4.6 - The ciphertext OIKYWVHBX was produced by...Ch. 4.6 - Express the Vigenère cipher as a cryptosystem. To...Ch. 4.6 - Prob. 21ECh. 4.6 - Once the length of the key string of a Vigènere...Ch. 4.6 - Prob. 23ECh. 4.6 - In Exercised 24-27 first express your answers...Ch. 4.6 - In Exercised 24-27 first express your answers...Ch. 4.6 - In Exercised 24-27 first express your answers...Ch. 4.6 - In Exercised 24-27 first express your answers...Ch. 4.6 - Suppose that (n, e) is an RSA encryption key, with...Ch. 4.6 - Describe the steps that Alice and Bob follow when...Ch. 4.6 - Describe the steps that Alice and Bob follow when...Ch. 4.6 - In Exercises 31-32 suppose that Alice and Bob have...Ch. 4.6 - In Exercises 31-32 suppose that Alice and Bob have...Ch. 4.6 - We describe a basis key exchange protocol using...Ch. 4.6 - Prob. 34ECh. 4.6 - Show that the Paillier cryptosystem is additively...Ch. 4 - Find 210 div 17 and 210 mod 17.Ch. 4 - a) Define what it means for a and b to be...Ch. 4 - Show if ab(modm) and cd(modm) , then a+cb+d(modm)...Ch. 4 - Describe a procedure for converting decimal (base...Ch. 4 - Prob. 5RQCh. 4 - Convert (7206)8 and (AOEB)16 to a binary...Ch. 4 - State the fundamental theorem of arithmetic.Ch. 4 - a) Describe a procedure for finding the prime...Ch. 4 - a) Define the greatest common divisor of two...Ch. 4 - a) How can you find a linear combination (with...Ch. 4 - a) What does it mean for a to be an inverse of a...Ch. 4 - a) How can an inverse of a modulo m be used to...Ch. 4 - a) State the Chinese remainder theorem. b) Find...Ch. 4 - Suppose that 2n11(modn) . Is n necessarily prime?Ch. 4 - Use Fermat’s little theorem to evaluate 9200 mod...Ch. 4 - Explain how the check digit is found for a 10-digi...Ch. 4 - Encrypt the meassage APPLES AND ORGANGES using a...Ch. 4 - a) What is the difference between a public key and...Ch. 4 - Explain how encryption and decryption are done in...Ch. 4 - Describe how two parties can share a secret key...Ch. 4 - The odometer on a car goes to up 100,000 miles....Ch. 4 - a) Explain why n div 7 equals the number of...Ch. 4 - Find four numbers congruent to 5 modulo 17.Ch. 4 - Show that if a and d are positive integers, then...Ch. 4 - Show that if acbc(modm) where a,b,c, and m are...Ch. 4 - Show that the sum of the squares of two odd...Ch. 4 - Show that if n2+1 is a perfect square, where n is...Ch. 4 - Prove that there are no solutions in integers x...Ch. 4 - Develop a test for divisibility of a positive...Ch. 4 - Develop a test for divisibility of a positive...Ch. 4 - Devise an algorithm for guessing a number between...Ch. 4 - Determine the complexity, in terms of the number...Ch. 4 - Show that an integer is divisible by 9 if and only...Ch. 4 - Show that a and b are positive irrational numbers...Ch. 4 - Prove there are infinitely many primes by showing...Ch. 4 - Find a positive integer n for which Qn=n!+1 is not...Ch. 4 - Use Dirichlet’s theorem, which states there are...Ch. 4 - Prove that if n is a positive integer such that...Ch. 4 - Show that every integer greater than 11 is the sum...Ch. 4 - Find the five smallest consecutive composite...Ch. 4 - Show that Goldbach’s conjecture, which states that...Ch. 4 - Find an arithmetic progression of length six...Ch. 4 - Prove that if f(x) is a nonconstant polynomial...Ch. 4 - How many zeros are at the end of the binary...Ch. 4 - Use the Euclidean algorithm to find the greatest...Ch. 4 - How many divisions are required to find gcd(144,...Ch. 4 - Find gcd(2n+1,3n+2) , where n is a positive...Ch. 4 - Show that if a and b are positive integers with ab...Ch. 4 - Adapt the proof that here are infinitely many...Ch. 4 - Explain why you cannot directly adapt the proof...Ch. 4 - Explain why you cannot directly adapt the proof...Ch. 4 - Show that if the smallest prime factor p of the...Ch. 4 - Prob. 33SECh. 4 - Find a set of four mutually relatively prime...Ch. 4 - For which positive integers n is n4+nn prime?Ch. 4 - Show that the system of congruences x2(mod6) and...Ch. 4 - Find all solutions of the system of congruences...Ch. 4 - a) Show that the system of congruences xa1(modm1)...Ch. 4 - Prove that 30 divisible n9-n for every nonnegative...Ch. 4 - Prove that n12-1 is divisible by 35 for every...Ch. 4 - Show that if p and q are distinct prime numbers,...Ch. 4 - Determine whether each of these 13-digit numbers...Ch. 4 - Show that the check digit of an ISBN-13 can always...Ch. 4 - Show that there are transpositions of two digit...Ch. 4 - Prob. 45SECh. 4 - Show that the check digit of an RTN can detect all...Ch. 4 - The encrypted version of message is LJMKG MG-MXF...Ch. 4 - Use the autokey cipher to encrypt the message NOW...Ch. 4 - Use the auto key cipher to encrypt the message THE...Ch. 4 - Given integers n and b, each greater than 1, find...Ch. 4 - Given the positive integers a, b, and m with m1 ,...Ch. 4 - Given a positive integer, find the cantor...Ch. 4 - Give a positive integer, determine whether it is...Ch. 4 - Given a positive integer, find the prime...Ch. 4 - Given two positive integers, find their greatest...Ch. 4 - Prob. 7CPCh. 4 - Prob. 8CPCh. 4 - Prob. 9CPCh. 4 - Given n liner congruences modulo pairwise...Ch. 4 - Prob. 11CPCh. 4 - Prob. 12CPCh. 4 - Prob. 13CPCh. 4 - Prob. 14CPCh. 4 - Prob. 15CPCh. 4 - Find the original plaintext message from the...Ch. 4 - Prob. 17CPCh. 4 - Prob. 18CPCh. 4 - Given a valid RSA key (n, e), and the primes p and...Ch. 4 - Given a message encrypted using the RSA...Ch. 4 - Generate a shared key using the Diffie-Hellman key...Ch. 4 - Prob. 22CPCh. 4 - Determine whether 2p1 is prime for each of the...Ch. 4 - Prob. 2CAECh. 4 - Prob. 3CAECh. 4 - Prob. 4CAECh. 4 - Prob. 5CAECh. 4 - Prob. 6CAECh. 4 - Prob. 7CAECh. 4 - Prob. 8CAECh. 4 - Prob. 9CAECh. 4 - Prob. 1WPCh. 4 - Explain how probabilistic primality tests are used...Ch. 4 - The question of whether there are infinitely many...Ch. 4 - Prob. 4WPCh. 4 - Describe the algorithms that are actually used by...Ch. 4 - Describe the history of the Chinese remainder...Ch. 4 - When are the numbers of a sequence truly random...Ch. 4 - Prob. 8WPCh. 4 - Prob. 9WPCh. 4 - Prob. 10WPCh. 4 - Prob. 11WPCh. 4 - Describe how public key cryptography can be used...Ch. 4 - Describe the Rabin public key cryptosystem,...Ch. 4 - Explain why it would be unsuitable to use p, where...Ch. 4 - Prob. 15WPCh. 4 - Explain the steps that Gentry used to construct a...
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- Roedel Electronics produces tablet computer accessories, including integrated keyboard tablet stands that connect a keyboard to a tablet device and holds the device at a preferred angle for easy viewing and typing. Roedel produces two sizes of integrated keyboard tablet stands, small and large. Each size uses the same keyboard attachment, but the stand consists of two different pieces, a top flap and a vertical stand that differ by size. Thus, a completed integrated keyboard tablet stand consists of three subassemblies that are manufactured by Roedel: a keyboard, a top flap, and a vertical stand. Roedel's sales forecast indicates that 7,000 small integrated keyboard tablet stands and 5,000 large integrated keyboard tablet stands will be needed to satisfy demand during the upcoming Christmas season. Because only 500 hours of in-house manufacturing time are available, Roedel is considering purchasing some, or all, of the subassemblies from outside suppliers. If Roedel manufactures a…arrow_forwardShow three different pairs of integers, a and b, where at least one example includes a negative integer. For each of your examples, determine if each of the following statements are true or falsearrow_forwardThe scores of 8 students on the midterm exam and final exam were as follows. Student Midterm Final Anderson 98 89 Bailey 88 74 Cruz 87 97 DeSana 85 79 Erickson 85 94 Francis 83 71 Gray 74 98 Harris 70 91 Find the value of the (Spearman's) rank correlation coefficient test statistic that would be used to test the claim of no correlation between midterm score and final exam score. Round your answer to 3 places after the decimal point, if necessary. Test statistic: rs =arrow_forward
- (a) Develop a model that minimizes semivariance for the Hauck Financial data given in the file HauckData with a required return of 10%. Assume that the five planning scenarios in the Hauck Financial rvices model are equally likely to occur. Hint: Modify model (8.10)-(8.19). Define a variable d, for each scenario and let d₂ > R - R¸ with d ≥ 0. Then make the objective function: Min Let FS = proportion of portfolio invested in the foreign stock mutual fund IB = proportion of portfolio invested in the intermediate-term bond fund LG = proportion of portfolio invested in the large-cap growth fund LV = proportion of portfolio invested in the large-cap value fund SG = proportion of portfolio invested in the small-cap growth fund SV = proportion of portfolio invested in the small-cap value fund R = the expected return of the portfolio R = the return of the portfolio in years. Min s.t. R₁ R₂ = R₁ R R5 = FS + IB + LG + LV + SG + SV = R₂ R d₁ =R- d₂z R- d₂ ZR- d₁R- d≥R- R = FS, IB, LG, LV, SG, SV…arrow_forwardThe Martin-Beck Company operates a plant in St. Louis with an annual capacity of 30,000 units. Product is shipped to regional distribution centers located in Boston, Atlanta, and Houston. Because of an anticipated increase in demand, Martin-Beck plans to increase capacity by constructing a new plant in one or more of the following cities: Detroit, Toledo, Denver, or Kansas. The following is a linear program used to determine which cities Martin-Beck should construct a plant in. Let y₁ = 1 if a plant is constructed in Detroit; 0 if not y₂ = 1 if a plant is constructed in Toledo; 0 if not y₂ = 1 if a plant is constructed in Denver; 0 if not y = 1 if a plant is constructed in Kansas City; 0 if not. The variables representing the amount shipped from each plant site to each distribution center are defined just as for a transportation problem. *,, = the units shipped in thousands from plant i to distribution center j i = 1 (Detroit), 2 (Toledo), 3 (Denver), 4 (Kansas City), 5 (St.Louis) and…arrow_forwardConsider the following mixed-integer linear program. Max 3x1 + 4x2 s.t. 4x1 + 7x2 ≤ 28 8x1 + 5x2 ≤ 40 x1, x2 ≥ and x1 integer (a) Graph the constraints for this problem. Indicate on your graph all feasible mixed-integer solutions. On the coordinate plane the horizontal axis is labeled x1 and the vertical axis is labeled x2. A region bounded by a series of connected line segments, and several horizontal lines are on the graph. The series of line segments connect the approximate points (0, 4), (3.889, 1.778), and (5, 0). The region is above the horizontal axis, to the right of the vertical axis, and below the line segments. At each integer value between 0 and 4 on the vertical axis, a horizontal line extends out from the vertical axis to the series of connect line segments. On the coordinate plane the horizontal axis is labeled x1 and the vertical axis is labeled x2. A region bounded by a series of connected line segments, and several…arrow_forward
- Consider the nonlinear optimization model stated below. Min s.t. 2x²-18x + 2XY + y² - 14Y + 53 x + 4Y ≤ 8 (a) Find the minimum solution to this problem. |at (X, Y) = (b) If the right-hand side of the constraint is increased from 8 to 9, how much do you expect the objective function to change? Based on the dual value on the constraint X + 4Y ≤ 8, we expect the optimal objective function value to decrease by (c) Resolve the problem with a new right-hand side of the constraint of 9. How does the actual change compare with your estimate? If we resolve the problem with a new right-hand-side of 9 the new optimal objective function value is| , so the actual change is a decrease of rather than what we expected in part (b).arrow_forwardStatement:If 2 | a and 3| a, then 6 a. So find three integers, and at least one integer should be negative. For each of your examples, determine if the statement is true or false.arrow_forwardStatement: If 4 | a and 6 | a, then 24 | a. So find three integers, and at least one integer should be negative. For each of your examples, determine if the statement is true or false.arrow_forward
- 2) dassify each critical point of the given plane autovers system x'=x-2x²-2xy y' = 4y-Sy³-7xyarrow_forwardEvaluate the next integralarrow_forward1. For each of the following, find the critical numbers of f, the intervals on which f is increasing or decreasing, and the relative maximum and minimum values of f. (a) f(x) = x² - 2x²+3 (b) f(x) = (x+1)5-5x-2 (c) f(x) = x2 x-9 2. For each of the following, find the intervals on which f is concave upward or downward and the inflection points of f. (a) f(x) = x - 2x²+3 (b) g(x) = x³- x (c) f(x)=x-6x3 + x-8 3. Find the relative maximum and minimum values of the following functions by using the Second Derivative Test. (a) f(x)=1+3x² - 2x3 (b) g(x) = 2x3 + 3x² - 12x-4arrow_forward
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