 4.1.1E: Does 17 divide each of these numbers?a) 68 b) 84 c) 357 d) 1001
 4.1.2E: Prove that if a is an integer other than 0. Thena) 1 divides a. b) ...
 4.1.3E: Prove that part (ii) of Theorem 1 is true.
 4.1.4E: Prove that part (iii) of Theorem 1 is true.
 4.1.5E: Show that if a  b and b  a. where a and b are integers, then a = ...
 4.1.6E: Show that if a, b, c, and d are integers, where a ? 0, such that a ...
 4.1.7E: Show that if a, b, and c arc integers, where a ? 0 and c ? 0, such ...
 4.1.8E: Prove or disprove that if abc, where a, b. and c are positive inte...
 4.1.9E: What are the quotient and remainder whena) 19 is divided by 7?_____...
 4.1.10E: What are the quotient and remainder whena) 44 is divided by 8?_____...
 4.1.11E: What time does a 12hour clock reada) 80 hours after it reads 11:00...
 4.1.12E: What time does a 24hour clock reada) 100 hours after it reads 2:00...
 4.1.13E: Suppose that a and b are integers, a ? 4 (mod 13), and b ? 9(mod 13...
 4.1.14E: Suppose that a and b are integers. a ? 11 (mod 19), and b ? 3 (mod ...
 4.1.15E: Let m be a positive integer. Show that a ? b (mod m) if a mod m = b...
 4.1.16E: Let m be a positive integer. Show that a mod m = b mod m if a ? b (...
 4.1.17E: Show that if n and k are positive integers, then
 4.1.18E: Show that if a is an integer and d is an integer greater than 1, th...
 4.1.19E: Find a formula for the integer with smallest absolute value that is...
 4.1.20E: Evaluate these quantities.a) 17 mod 2________________b) 144 mod 7_...
 4.1.21E: Evaluate these quantities.a) 13 mod 3________________b) 97 mod 11_...
 4.1.22E: Find a div m and a mod in whena) a = 111, m =99.________________b)...
 4.1.23E: Find a div m and a mod m whena) a = 228, m = 119.________________b)...
 4.1.24E: Find the integer a such thata) a ? 43 (mod 23) and 22 ? a ? 0.____...
 4.1.25E: Find the integer a such thata) a ?15 (mod 27) and 26 ? a ? 0.____...
 4.1.26E: List five integers that are congruent to 4 modulo 12.
 4.1.27E: List all integers between 100 and 100 that are congruent to 1 mod...
 4.1.28E: Decide whether each of these integers is congruent to 3 modulo 7.a)...
 4.1.29E: Decide whether each of these integers is congruent to 5 modulo 17.a...
 4.1.30E: Find each of these values.a) (177 mod 31 + 270 mod 31) mod 31______...
 4.1.31E: Find each of these values.a) ( 133 mod 23 + 261 mod 23) mod 23____...
 4.1.32E: Find each of these values.a) (192 mod 41) mod 9________________b) (...
 4.1.33E: Find each of these values.a) (992 mod 32)3 mod 15________________b)...
 4.1.34E: Show that if a ? b (mod m) and c ? d (mod m), where a, b, c. d, and...
 4.1.35E: Show that if n  m. where n and m arc integers greater than 1, and ...
 4.1.36E: Show that if a, b, c, and m are integers such that m ? 2, c > 0, an...
 4.1.37E: Find counterexamples to each of these statements about congruences....
 4.1.38E: Show that if n is an integer then n2 ? 0 or 1 (mod 4).
 4.1.39E: Use Exercise 38 to show that if m is a positive integer of the form...
 4.1.40E: Prove that if n is an odd positive integer, then n2 = 1 (mod 8).
 4.1.41E: Show that if a. b, k. and m are integers such that k ? 1, m ? 2, an...
 4.1.42E: Show that Zm with addition modulo m, where m ? 2 is an integer, sat...
 4.1.43E: Show that Zm with multiplication modulo in. where m ? 2 is an integ...
 4.1.44E: Show that the distributive property of multiplication over addition...
 4.1.45E: Write out the addition and multiplication tables for Z5 (where by a...
 4.1.46E: Write out the addition and multiplication tables for Z6 (where by a...
 4.1.47E: Determine whether each of the functions f(a) = a div d and g(a) = a...
Solutions for Chapter 4.1: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 4.1
Get Full SolutionsThis textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Since 47 problems in chapter 4.1 have been answered, more than 123764 students have viewed full stepbystep solutions from this chapter. Chapter 4.1 includes 47 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Iterative method.
A sequence of steps intended to approach the desired solution.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!