 0.1CE: This software checks the validity of a Postal Service money order n...
 0.1E: For n = 5, 8, 12, 20, and 25, find all positive integers less than ...
 0.2CE: This software checks the validity of a UPC number. Use it to verify...
 0.2E: Determine gcd(24 ? 32 ?5 ? 72, 2 ?33 ? 7 ? 11) and lcm(23 ? 32? 5, ...
 0.3CE: This software checks the validity of a UPS number. Use it to verify...
 0.3E: Determine 51 mod 13, 342 mod 85, 62 mod 15, 10 mod 15, (82 ? 73) mo...
 0.4CE: This software checks the validity of an identification number on a ...
 0.4E: Find integers s and t such that 1 = 7 ? s + 11 ? t. Show that s and...
 0.5CE: Exercise 5. This software calculates the ISBN check digit. Use it t...
 0.5E: Show that if a and b are positive integers, then ab = lcm(a, b) ? g...
 0.6CE: Exercise 5. This software calculates the ISBN check digit. Use it t...
 0.6E: Suppose a and b are integers that divide the integer c. If a and b ...
 0.7E: If a and b are integers and n is a positive integer, prove that a m...
 0.8E: Let d = gcd(a, b). If a = da' and b = db', show that gcd(a', b') = 1.
 0.9E: Let n be a fixed positive integer greater than 1. If a mod n = a' a...
 0.10E: Let a and b be positive integers and let d = gcd(a, b) and m = lcm(...
 0.11E: Let n and a be positive integers and let d = gcd(a, n). Show that t...
 0.12E: Show that 5n + 3 and 7n + 4 are relatively prime for all n.
 0.13E: Suppose that m and n are relatively prime and r is any integer. Sho...
 0.14E: Let p, q, and r be primes other than 3. Show that 3 divides p2 + q2...
 0.15E: Prove that every prime greater than 3 can be written in the form 6n...
 0.16E: Determine 71000 mod 6 and 61001 mod 7.
 0.17E: Let a, b, s, and t be integers. If a mod st = b mod st, show that a...
 0.18E: Determine 8402 mod 5
 0.19E: Show that gcd(a, bc) = 1 if and only if gcd(a, b) = 1 and gcd(a, c)...
 0.20E: Let p1, p2, . . . , pn be primes. Show that p1 p2??? pn + 1 is divi...
 0.21E: Prove that there are infinitely many primes. (Hint: Use Exercise 20...
 0.22E: Express (–7 – 3i)–1 in standard form.
 0.23E: Express in standard form.
 0.24E: Express (cos 3600 + i sin 3600)1/8 in standard form without trig ex...
 0.25E: Prove that for any positive integer n, .
 0.26E: For every positive integer n, prove that 1 + 2 + ??? + n = n(n + 1)/2
 0.27E: For every positive integer n, prove that a set with exactly n eleme...
 0.28E: Prove that 2n32n –1 is always divisible by 17.
 0.29E: Prove that there is some positive integer n such that n, n + 1, n +...
 0.30E: (Generalized Euclid’s Lemma) If p is a prime and p divides a1a2?? ?...
 0.31E: Use the Generalized Euclid’s Lemma (see Exercise 30) to establish t...
 0.32E: What is the largest bet that cannot be made with chips worth $7.00 ...
 0.33E: Prove that the First Principle of Mathematical Induction is a conse...
 0.34E: The Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . . In ...
 0.35E: Prove by induction on n that for all positive integers n, n3 1 (n +...
 0.36E: Suppose that there is a statement involving a positive integer para...
 0.37E: In the cut “As” from Songs in the Key of Life, Stevie Wonder mentio...
 0.38E: Prove that for every integer n, n3 mod 6 = n mod 6.
 0.39E: If it is 2:00 a.m. now, what time will it be 3736 hours from now?
 0.40E: Determine the check digit for a money order with identification num...
 0.41E: Suppose that in one of the noncheck positions of a money order numb...
 0.42E: Suppose that a money order identification number and check digit of...
 0.43E: A transposition error involving distinct adjacent digits is one of ...
 0.44E: Determine the check digit for the Avis rental car with identificati...
 0.45E: Show that a substitution of a digit ai ' for the digit ai (ai ' ? a...
 0.46E: Determine which transposition errors involving adjacent digits are ...
 0.47E: Use the UPC scheme to determine the check digit for the number 0731...
 0.48E: Explain why the check digit for a money order for the number N is t...
 0.49E: The 10digit International Standard Book Number (ISBN10) a1a2a3a4a...
 0.50E: Suppose that an ISBN10 has a smudged entry where the question mark...
 0.51E: Suppose three consecutive digits abc of an ISBN10 are scrambled as...
 0.52E: The ISBN10 0669039254 is the result of a transposition of two a...
 0.53E: Suppose the weighting vector for ISBN10s were changed to (1, 2, 3,...
 0.54E: Use the twocheckdigit errorcorrection method described in this c...
 0.55E: Suppose that an eightdigit number has two check digits appended us...
 0.56E: The state of Utah appends a ninth digit a9 to an eightdigit driver...
 0.57E: Complete the proof of Theorem 0.8.Reference:
 0.58E: Let S be the set of real numbers. If a, b ? S, define a ~ b if a – ...
 0.59E: Let S be the set of integers. If a, b ? S, define aRb if ab ?0. Is ...
 0.60E: Let S be the set of integers. If a, b ? S, define aRb if a +b is ev...
 0.61E: Complete the proof of Theorem 0.7 by showing that , is an equivalen...
 0.62E: Prove that 3, 5, and 7 are the only three consecutive odd integers ...
 0.63E: What is the last digit of 3100? What is the last digit of 2100?
 0.64E: Prove that none of the integers 11, 111, 1111, 11111, . . . is a sq...
 0.65E: (Cancellation Property) Suppose ?, ?, and ? are functions. If ?? = ...
Solutions for Chapter 0: Contemporary Abstract Algebra 8th Edition
Full solutions for Contemporary Abstract Algebra  8th Edition
ISBN: 9781133599708
Solutions for Chapter 0
Get Full SolutionsThis textbook survival guide was created for the textbook: Contemporary Abstract Algebra , edition: 8. Since 71 problems in chapter 0 have been answered, more than 46498 students have viewed full stepbystep solutions from this chapter. Contemporary Abstract Algebra was written by and is associated to the ISBN: 9781133599708. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 0 includes 71 full stepbystep solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.