Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN: 9781285463230
Author: Gilbert, Linda, Jimmie
Publisher: Cengage Learning,
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Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba…
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
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Chapter 2.8, Problem 26E

(a)

To determine

Each of the following ϕ(n).

(b)

To determine

To prove: ϕ(pj) = pj−1( p − 1 )

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Chapter 2 Solutions

Elements Of Modern Algebra

Ch. 2.1 - True or False Label each of the following...Ch. 2.1 - True or False Label each of the following...Ch. 2.1 - True or False Label each of the following...Ch. 2.1 - True or False Label each of the following...Ch. 2.1 - True or False Label each of the following...Ch. 2.1 - Prob. 6TFECh. 2.1 - Prob. 7TFECh. 2.1 - Prob. 8TFECh. 2.1 - Prob. 9TFECh. 2.1 - Prob. 10TFE
Ch. 2.1 - Prove that the equalities in Exercises 111 hold...Ch. 2.1 - Prob. 2ECh. 2.1 - Prob. 3ECh. 2.1 - Prob. 4ECh. 2.1 - Prove that the equalities in Exercises hold for...Ch. 2.1 - Prob. 6ECh. 2.1 - Prob. 7ECh. 2.1 - Prove that the equalities in Exercises hold for...Ch. 2.1 - Prove that the equalities in Exercises hold for...Ch. 2.1 - Prob. 10ECh. 2.1 - Prob. 11ECh. 2.1 - Let A be a set of integers closed under...Ch. 2.1 - Prob. 13ECh. 2.1 - In Exercises , prove the statements concerning the...Ch. 2.1 - Prob. 15ECh. 2.1 - Prob. 16ECh. 2.1 - In Exercises , prove the statements concerning the...Ch. 2.1 - In Exercises , prove the statements concerning the...Ch. 2.1 - In Exercises 13-24, prove the statements...Ch. 2.1 - In Exercises 1324, prove the statements concerning...Ch. 2.1 - Prob. 21ECh. 2.1 - Prob. 22ECh. 2.1 - Prob. 23ECh. 2.1 - Prob. 24ECh. 2.1 - 25. Prove that if and are integers and, then...Ch. 2.1 - Prove that the cancellation law for multiplication...Ch. 2.1 - Let x and y be in Z, not both zero, then x2+y2Z+.Ch. 2.1 - Prob. 28ECh. 2.1 - Prob. 29ECh. 2.1 - Prob. 30ECh. 2.1 - 31. Prove that if is positive and is negative,...Ch. 2.1 - 32. Prove that if is positive and is positive,...Ch. 2.1 - 33. Prove that if is positive and is negative,...Ch. 2.1 - Prob. 34ECh. 2.1 - Prob. 35ECh. 2.2 - Prove that the statements in Exercises are true...Ch. 2.2 - Prove that the statements in Exercises are true...Ch. 2.2 - Prob. 3ECh. 2.2 - Prove that the statements in Exercises are true...Ch. 2.2 - Prob. 5ECh. 2.2 - Prob. 6ECh. 2.2 - Prob. 7ECh. 2.2 - Prob. 8ECh. 2.2 - Prob. 9ECh. 2.2 - Prob. 10ECh. 2.2 - Prob. 11ECh. 2.2 - Prob. 12ECh. 2.2 - Prob. 13ECh. 2.2 - Prob. 14ECh. 2.2 - Prob. 15ECh. 2.2 - Prove that the statements in Exercises 116 are...Ch. 2.2 - 17. Use mathematical induction to prove that the...Ch. 2.2 - Let be integers, and let be positive integers....Ch. 2.2 - Let xandy be integers, and let mandn be positive...Ch. 2.2 - Let xandy be integers, and let mandn be positive...Ch. 2.2 - Let x and y be integers, and let m and n be...Ch. 2.2 - Let x and y be integers, and let m and n be...Ch. 2.2 - Let and be integers, and let and be positive...Ch. 2.2 - Prob. 24ECh. 2.2 - Prob. 25ECh. 2.2 - Prob. 26ECh. 2.2 - Use the equation (nr1)+(nr)=(n+1r) for 1rn. And...Ch. 2.2 - Use the equation. (nr1)+(nr)=(n+1r) for 1rn....Ch. 2.2 - Prob. 29ECh. 2.2 - Prob. 30ECh. 2.2 - Prob. 31ECh. 2.2 - In Exercise use mathematical induction to prove...Ch. 2.2 - In Exercise 3236 use mathematical induction to...Ch. 2.2 - Prob. 34ECh. 2.2 - Prob. 35ECh. 2.2 - Prob. 36ECh. 2.2 - Prob. 37ECh. 2.2 - Prob. 38ECh. 2.2 - Prob. 39ECh. 2.2 - Exercise can be generalized as follows: If and...Ch. 2.2 - Prob. 41ECh. 2.2 - Prob. 42ECh. 2.2 - In Exercise , use generalized induction to prove...Ch. 2.2 - Prob. 44ECh. 2.2 - In Exercise 4145, use generalized induction to...Ch. 2.2 - Use generalized induction and Exercise 43 to prove...Ch. 2.2 - Use generalized induction and Exercise 43 to prove...Ch. 2.2 - Assume the statement from Exercise 30 in section...Ch. 2.2 - Show that if the statement is assumed to be true...Ch. 2.2 - Show that if the statement 1+2+3+...+n=n(n+1)2+2...Ch. 2.2 - Given the recursively defined sequence a1=1,a2=4,...Ch. 2.2 - Given the recursively defined sequence...Ch. 2.2 - Given the recursively defined sequence a1=0,a2=30,...Ch. 2.2 - Given the recursively defined sequence , and , use...Ch. 2.2 - The Fibonacci sequence fn=1,1,2,3,5,8,13,21,... is...Ch. 2.2 - Let f1,f2,...,fn be permutations on a nonempty set...Ch. 2.2 - Define powers of a permutation on by the...Ch. 2.3 - Label each of the following statements as either...Ch. 2.3 - Label each of the following statement as either...Ch. 2.3 - Label each of the following statement as either...Ch. 2.3 - Label each of the following statement as either...Ch. 2.3 - Label each of the following statement as either...Ch. 2.3 - Label each of the following statement as either...Ch. 2.3 - Prob. 7TFECh. 2.3 - Prob. 8TFECh. 2.3 - Label each of the following statement as either...Ch. 2.3 - Prob. 10TFECh. 2.3 - Prob. 1ECh. 2.3 - Prob. 2ECh. 2.3 - Write and as given in Exercises, find the q and...Ch. 2.3 - Write a and b as given in Exercises 316, find the...Ch. 2.3 - Write and as given in Exercises, find the q and...Ch. 2.3 - Prob. 6ECh. 2.3 - Prob. 7ECh. 2.3 - Prob. 8ECh. 2.3 - Prob. 9ECh. 2.3 - Write a and b as given in Exercises 316, find the...Ch. 2.3 - Write a and b as given in Exercises 316, find the...Ch. 2.3 - Write and as given in Exercises, find the and ...Ch. 2.3 - Write and as given in Exercises, find the and ...Ch. 2.3 - Write and as given in Exercises, find the and ...Ch. 2.3 - Write and as given in Exercises, find the and...Ch. 2.3 - Write a and b as given in Exercises 316, find the...Ch. 2.3 - 17. If a,b and c are integers such that ab and ac,...Ch. 2.3 - Let R be the relation defined on the set of...Ch. 2.3 - 19. If and are integers with and . Prove that...Ch. 2.3 - Let a,b,c and d be integers such that ab and cd....Ch. 2.3 - Prove that if and are integers such that and ,...Ch. 2.3 - Prove that if and are integers such that and ,...Ch. 2.3 - Let a and b be integers such that ab and ba. Prove...Ch. 2.3 - Let , and be integers . Prove or disprove that ...Ch. 2.3 - Let ,, and be integers. Prove or disprove that ...Ch. 2.3 - 26. Let be an integer. Prove that . (Hint:...Ch. 2.3 - Let a be an integer. Prove that 3|a(a+1)(a+2)....Ch. 2.3 - Let a be an odd integer. Prove that 8|(a21).Ch. 2.3 - Prob. 29ECh. 2.3 - Let be as described in the proof of Theorem. Give...Ch. 2.3 - Prob. 31ECh. 2.3 - Prob. 32ECh. 2.3 - Prob. 33ECh. 2.3 - Prob. 34ECh. 2.3 - Prob. 35ECh. 2.3 - Prob. 36ECh. 2.3 - In Exercises, use mathematical induction to prove...Ch. 2.3 - Prob. 38ECh. 2.3 - Prob. 39ECh. 2.3 - In Exercises, use mathematical induction to prove...Ch. 2.3 - In Exercises, use mathematical induction to prove...Ch. 2.3 - Prob. 42ECh. 2.3 - Prob. 43ECh. 2.3 - Prob. 44ECh. 2.3 - Prob. 45ECh. 2.3 - In Exercises, use mathematical induction to prove...Ch. 2.3 - Prob. 47ECh. 2.3 - Prob. 48ECh. 2.3 - 49. a. The binomial coefficients are defined in...Ch. 2.4 - True or false Label each of the following...Ch. 2.4 - True or false Label each of the following...Ch. 2.4 - True or false Label each of the following...Ch. 2.4 - True or false Label each of the following...Ch. 2.4 - True or false Label each of the following...Ch. 2.4 - True or false Label each of the following...Ch. 2.4 - True or false Label each of the following...Ch. 2.4 - Prob. 8TFECh. 2.4 - Prob. 9TFECh. 2.4 - Prob. 10TFECh. 2.4 - True or false Label each of the following...Ch. 2.4 - True or false Label each of the following...Ch. 2.4 - True or false Label each of the following...Ch. 2.4 - List all the primes lessthan 100.Ch. 2.4 - For each of the following pairs, write andin...Ch. 2.4 - In each part, find the greatest common divisor...Ch. 2.4 - Find the smallest integer in the given set. { and ...Ch. 2.4 - Prove that if p and q are distinct primes, then...Ch. 2.4 - Show that n2n+5 is a prime integer when n=1,2,3,4...Ch. 2.4 - If a0 and ab, then prove or disprove that (a,b)=a.Ch. 2.4 - If , prove . Ch. 2.4 - Let , and be integers such that . Prove that if ,...Ch. 2.4 - Let be a nonzero integer and a positive integer....Ch. 2.4 - Let ac and bc, and (a,b)=1, prove that ab divides...Ch. 2.4 - Prove that if , , and , then . Ch. 2.4 - Let and . Prove or disprove that . Ch. 2.4 - Prob. 14ECh. 2.4 - Let r0=b0. With the notation used in the...Ch. 2.4 - Prob. 16ECh. 2.4 - Prob. 17ECh. 2.4 - Prob. 18ECh. 2.4 - Prove that if n is a positive integer greater than...Ch. 2.4 - Prob. 20ECh. 2.4 - Let (a,b)=1 and (a,c)=1. Prove or disprove that...Ch. 2.4 - Prob. 22ECh. 2.4 - Prob. 23ECh. 2.4 - Let (a,b)=1. Prove that (a,bn)=1 for all positive...Ch. 2.4 - Prove that if m0 and (a,b) exists, then...Ch. 2.4 - Prove that if d=(a,b), a=a0d, and b=b0d, then...Ch. 2.4 - Prove that the least common multiple of two...Ch. 2.4 - Let and be positive integers. If and is the...Ch. 2.4 - Prob. 29ECh. 2.4 - Let , and be three nonzero integers. Use...Ch. 2.4 - Find the greatest common divisor of a,b, and c and...Ch. 2.4 - Use the second principle of Finite Induction to...Ch. 2.4 - Use the fact that 3 is a prime to prove that there...Ch. 2.4 - Prob. 34ECh. 2.4 - Prove that 23 is not a rational number.Ch. 2.5 - True or False Label each of the following...Ch. 2.5 - True or False Label each of the following...Ch. 2.5 - Label each of the following statements as either...Ch. 2.5 - Label each of the following statements as either...Ch. 2.5 - Label each of the following statements as either...Ch. 2.5 - Label each of the following statements as either...Ch. 2.5 - Label each of the following statements as either...Ch. 2.5 - In this exercise set, all variables are...Ch. 2.5 - In this exercise set, all variables are...Ch. 2.5 - Find a solution , , for each of the congruences ...Ch. 2.5 - Find a solution , , for each of the congruences ...Ch. 2.5 - Find a solution x, 0xn, for each of the...Ch. 2.5 - Prob. 6ECh. 2.5 - Find a solution x, 0xn, for each of the...Ch. 2.5 - Find a solution x, 0xn, for each of the...Ch. 2.5 - Find a solution , , for each of the congruences ...Ch. 2.5 - Prob. 10ECh. 2.5 - Find a solution , , for each of the congruences ...Ch. 2.5 - Prob. 12ECh. 2.5 - Find a solution x, 0xn, for each of the...Ch. 2.5 - Prob. 14ECh. 2.5 - Find a solution x, 0xn, for each of the...Ch. 2.5 - Prob. 16ECh. 2.5 - Find a solution x, 0xn, for each of the...Ch. 2.5 - Find a solution x, 0xn, for each of the...Ch. 2.5 - Find a solution x, 0xn, for each of the...Ch. 2.5 - Prob. 20ECh. 2.5 - Prob. 21ECh. 2.5 - Prob. 22ECh. 2.5 - Prob. 23ECh. 2.5 - Find a solution , , for each of the congruences ...Ch. 2.5 - 25. Complete the proof of Theorem : If and is...Ch. 2.5 - Complete the proof of Theorem 2.24: If ab(modn)...Ch. 2.5 - Prove that if a+xa+y(modn), then xy(modn).Ch. 2.5 - 28. If and where , prove that . Ch. 2.5 - 29. Find the least positive integer that is...Ch. 2.5 - 30. Prove that any positive integer is congruent...Ch. 2.5 - 31. If , prove that for every positive integer . Ch. 2.5 - 32. Prove that if is an integer, then either or...Ch. 2.5 - Prove or disprove that if n is odd, then...Ch. 2.5 - Prob. 34ECh. 2.5 - Prob. 35ECh. 2.5 - Prob. 36ECh. 2.5 - Prob. 37ECh. 2.5 - Prob. 38ECh. 2.5 - Prob. 39ECh. 2.5 - In the congruences axb(modn) in Exercises 4053, a...Ch. 2.5 - In the congruences in Exercises, and may not be...Ch. 2.5 - In the congruences in Exercises, and may not be...Ch. 2.5 - In the congruences axb(modn) in Exercises 4053, a...Ch. 2.5 - In the congruences in Exercises, and may not be...Ch. 2.5 - Prob. 45ECh. 2.5 - In the congruences in Exercises, and may not be...Ch. 2.5 - Prob. 47ECh. 2.5 - Prob. 48ECh. 2.5 - In the congruences in Exercises, and may not be...Ch. 2.5 - In the congruences in Exercises, and may not be...Ch. 2.5 - In the congruences ax b (mod n) in Exercises...Ch. 2.5 - In the congruences axb(modn) in Exercises 4053, a...Ch. 2.5 - Prob. 53ECh. 2.5 - 54. Let be a prime integer. Prove Fermat's Little...Ch. 2.5 - 55. Prove the Chinese Remainder Theorem: Let , , ....Ch. 2.5 - 56. Solve the following systems of congruences. ...Ch. 2.5 - Prob. 57ECh. 2.5 - a. Prove that 10n(1)n(mod11) for every positive...Ch. 2.6 - Label each of the following statements as either...Ch. 2.6 - True or False Label each of the following...Ch. 2.6 - Prob. 3TFECh. 2.6 - True or False Label each of the following...Ch. 2.6 - True or False Label each of the following...Ch. 2.6 - Prob. 6TFECh. 2.6 - Prob. 7TFECh. 2.6 - Prob. 8TFECh. 2.6 - Prob. 1ECh. 2.6 - a. Verify that [ 1 ][ 2 ][ 3 ][ 4 ]=[ 4 ] in 5. b....Ch. 2.6 - Make addition tables for each of the following....Ch. 2.6 - Make multiplication tables for each of the...Ch. 2.6 - Find the multiplicative inverse of each given...Ch. 2.6 - Prob. 6ECh. 2.6 - Find all zero divisors in each of the following n....Ch. 2.6 - Whenever possible, find a solution for each of the...Ch. 2.6 - Let [ a ] be an element of n that has a...Ch. 2.6 - Solve each of the following equations by finding [...Ch. 2.6 - In Exercise, Solve the systems of equations in. ...Ch. 2.6 - In Exercise, Solve the systems of equations...Ch. 2.6 - In Exercise 1114, Solve the systems of equations...Ch. 2.6 - Prob. 14ECh. 2.6 - Prove Theorem. Theorem 2.30 Multiplication...Ch. 2.6 - Prob. 16ECh. 2.6 - Prob. 17ECh. 2.6 - Prob. 18ECh. 2.6 - Prob. 19ECh. 2.6 - Prob. 20ECh. 2.6 - Prob. 21ECh. 2.6 - Prob. 22ECh. 2.6 - Prob. 23ECh. 2.6 - Prob. 24ECh. 2.6 - Prob. 25ECh. 2.6 - Prove that a nonzero element in is a zero divisor...Ch. 2.7 - True or False Label each of the following...Ch. 2.7 - Prob. 2TFECh. 2.7 - Prob. 3TFECh. 2.7 - Prob. 4TFECh. 2.7 - Suppose 4- bit words abcd are mapped onto 5- bit...Ch. 2.7 - Prob. 2ECh. 2.7 - Prob. 3ECh. 2.7 - Prob. 4ECh. 2.7 - Suppose a codding scheme is devised that maps -bit...Ch. 2.7 - Suppose the probability of erroneously...Ch. 2.7 - Prob. 7ECh. 2.7 - Suppose the probability of incorrectly...Ch. 2.7 - Prob. 9ECh. 2.7 - Is the identification number 11257402 correct if...Ch. 2.7 - Show that the check digit in bank identification...Ch. 2.7 - Suppose that the check digit is computed as...Ch. 2.7 - Prob. 13ECh. 2.7 - Prob. 14ECh. 2.7 - Verify that the check digit in a UPC symbol...Ch. 2.7 - Prob. 16ECh. 2.7 - Prob. 17ECh. 2.7 - Prob. 18ECh. 2.7 - Prob. 19ECh. 2.7 - Prob. 20ECh. 2.7 - Prob. 21ECh. 2.7 - Prob. 22ECh. 2.7 - Prob. 23ECh. 2.7 - Prob. 24ECh. 2.7 - Prob. 25ECh. 2.7 - Prob. 26ECh. 2.8 - Label each of the following statements as either...Ch. 2.8 - Prob. 2TFECh. 2.8 - Prob. 3TFECh. 2.8 - In the -letter alphabet A described in Example,...Ch. 2.8 - Prob. 2ECh. 2.8 - Prob. 3ECh. 2.8 - Prob. 4ECh. 2.8 - In the -letter alphabet described in Example, use...Ch. 2.8 - Prob. 6ECh. 2.8 - Prob. 7ECh. 2.8 - Use the alphabet C from the preceding problem and...Ch. 2.8 - Suppose that in a long ciphertext message the...Ch. 2.8 - Suppose that in a long ciphertext message the...Ch. 2.8 - Suppose the alphabet consists of a through z, in...Ch. 2.8 - Suppose the alphabet consists of a through, in...Ch. 2.8 - Prob. 13ECh. 2.8 - Prob. 14ECh. 2.8 - a. Excluding the identity cipher, how many...Ch. 2.8 - Rework Example 5 by breaking the message into...Ch. 2.8 - Suppose that in an RSA Public Key Cryptosystem,...Ch. 2.8 - Suppose that in an RSA Public Key Cryptosystem,...Ch. 2.8 - Suppose that in an RSA Public Key Cryptosystem....Ch. 2.8 - Suppose that in an RSA Public Key Cryptosystem....Ch. 2.8 - Prob. 21ECh. 2.8 - Prob. 22ECh. 2.8 - Prob. 23ECh. 2.8 - Prob. 24ECh. 2.8 - Prob. 25ECh. 2.8 - Prob. 26E
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