Discrete Mathematics and Its Applications
8th Edition
ISBN: 9781260501759
Author: ROSEN
Publisher: MCG
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Textbook Question
Chapter 4, Problem 34SE
Find a set of four mutually relatively prime integers such that no two of them are relatively prime.
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Chapter 4 Solutions
Discrete Mathematics and Its Applications
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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- Given sets X and Y and Z, can you prove that (X-(Y u Z)) u (Y-(X u Z)) is a subset of (X u Y) - (X intersection Y)arrow_forward4. A car travels in a straight line for one hour. Its velocity, v, in miles per hour at six minute intervals is shown in the table. For each problem, approximate the distance the car traveled (in miles) using the given method, on the provided interval, and with the given number of rectangles or trapezoids, n. Time (min) 0 6 12 18|24|30|36|42|48|54|60 Speed (mph) 0 10 20 40 60 50 40 30 40 40 65 a.) Left Rectangles, [0, 30] n=5 b.) Right Rectangles, [24, 42] n=3 c.) Midpoint Rectangles, [24, 60] n=3 d.) Trapezoids, [0, 24] n=4arrow_forwardGiven the functions A and B, can you prove that if B ◦ A is bijective, then A is injective and B is surjectivearrow_forward
- - + ++ Table 2: Crack Experiment for Exercise 2 A B C D Treatment Combination (1) Replicate I II 7.037 6.376 14.707 15.219 |++++ 1 བྱ॰༤༠སྦྱོ སྦྱོཋཏྟཱུ a b ab 11.635 12.089 17.273 17.815 с ас 10.403 10.151 4.368 4.098 bc abc 9.360 9.253 13.440 12.923 d 8.561 8.951 ad 16.867 17.052 bd 13.876 13.658 abd 19.824 19.639 cd 11.846 12.337 acd 6.125 5.904 bcd 11.190 10.935 abcd 15.653 15.053 Question 3 Continuation of Exercise 2. One of the variables in the experiment described in Exercise 2, heat treatment method (C), is a categorical variable. Assume that the remaining factors are continuous. (a) Write two regression models for predicting crack length, one for each level of the heat treatment method variable. What differences, if any, do you notice in these two equations? (b) Generate appropriate response surface contour plots for the two regression models in part (a). (c) What set of conditions would you recommend for the factors A, B, and D if you use heat treatment method C = +? (d) Repeat…arrow_forwardTerry has a square plot of land measuring 500 meters by 500 meters. She divided the land into 25 100-m by 100-m plots and created three raster maps showing the type of mineral, fruit tree, and energy available on each plot. Use the maps below to shade the blank maps according to each problem.arrow_forwardThe bracket BCD is hinged at C and attached to a control cable at B. Let F₁ = 275 N and F2 = 275 N. F1 B a=0.18 m C A 0.4 m -0.4 m- 0.24 m Determine the reaction at C. The reaction at C N Z F2 Darrow_forward
- Question 2 A nickel-titanium alloy is used to make components for jet turbine aircraft engines. Cracking is a potentially serious problem in the final part because it can lead to nonrecoverable failure. A test is run at the parts producer to determine the effect of four factors on cracks. The four factors are: pouring temperature (A), titanium content (B), heat treatment method (C), amount of grain refiner used (D). Two replicates of a 24 design are run, and the length of crack (in mm x10-2) induced in a sample coupon subjected to a standard test is measured. The data are shown in Table 2. 1 (a) Estimate the factor effects. Which factor effects appear to be large? (b) Conduct an analysis of variance. Do any of the factors affect cracking? Use a = 0.05. (c) Write down a regression model that can be used to predict crack length as a function of the significant main effects and interactions you have identified in part (b). (d) Analyze the residuals from this experiment. (e) Is there an…arrow_forwardShow the stepsarrow_forwardThe correct answer is C,i know that we need to use stokes theorem and parametrize the equations then write the equation F with respect to the curve but i cant seem to find a way to do it, the integral should be from 0 to 2pi but i might be wrongcould you show me the steps to get to 18piarrow_forward
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