 5.6.1: 5 = 11 + 14. Show by computing each expression that 25 mod 6 = (11 ...
 5.6.2: 395 = 129 + 266. Show by computing each expression that 395 mod 4 =...
 5.6.3: 262 = 74 + 188. Show by computing each expression that 262 mod 13 =...
 5.6.4: Prove that for any integers x and y, (x + y) mod n = (x mod n + y m...
 5.6.5: 486 = 18 # 27. Show by computing each expression that 486 mod 5 = (...
 5.6.6: 7067 = 191 # 37. Show by computing each expression that 7067 mod 8 ...
 5.6.7: Prove that for any integers x and y, (x # y) mod n = (x mod n # y m...
 5.6.8: Prove that for any integers x and y, (x # y) mod n = (x mod n # y m...
 5.6.9: Using the hash function of Example 51, which of the following Socia...
 5.6.10: Find a set of five numbers in the range 30, 2004 that cause 100% co...
 5.6.11: Using a hash table of size 11 and the hash function x mod 11, show ...
 5.6.12: Using the completed hash table from Exercise 11, compute the averag...
 5.6.13: When a computer program is compiled, the compiler builds a symbol t...
 5.6.14: Explain what problem can arise if an item stored in a hash table is...
 5.6.15: A disadvantage of hashing with linear probing for collision resolut...
 5.6.16: Generalize the answers to Exercise 15 to explain why using linear p...
 5.6.17: Decode the following ciphertext messages that were encoded using a ...
 5.6.18: Using a Caesar cipher, encode the following plaintext messages. a. ...
 5.6.19: The following ciphertext message is intercepted; you suspect it is ...
 5.6.20: The following ciphertext message is intercepted; you suspect it is ...
 5.6.21: Use the algorithm of Example 53 to compute the 1bit circular left ...
 5.6.22: Use the algorithm of Example 53 to compute the 1bit circular left ...
 5.6.23: Consider a short form of DES that uses 16bit keys. Given the 16bi...
 5.6.24: Describe how to perform a 1bit right circular shift on a 4bit bin...
 5.6.25: Using RSA encryption/decryption, let p = 5 and q = 3. Then n = 15 a...
 5.6.26: Why is the RSA encryption of Exercise 25 a poor choice?
 5.6.27: Using RSA encryption/decryption, let p = 5 and q = 11. Then n = 55 ...
 5.6.28: Using RSA encryption/decryption, let p = 23 and q = 31. Then n = 71...
 5.6.29: a. All n people in a group wish to communicate with each other usin...
 5.6.30: Computer users are notoriously lax about choosing passwords; left t...
 5.6.31: a. The ISBN10 of the sixth edition of this book is 071676864C w...
 5.6.32: A bookstore placed an order for 2000 copies of Harry Potter and the...
 5.6.33: Given the 11 digits 02724911637, compute the check digit for the UP...
 5.6.34: A taxpayer wants his tax refund to be deposited directly to his ban...
 5.6.35: a. Write an algorithm to decompose a fourdigit integer into the on...
 5.6.36: The following quilt image is based on addition modulo n for what va...
 5.6.37: Prove that if x y (mod n) and c is a constant integer, then xc yc (...
 5.6.38: This exercise explores the converse of Exercise 37, which is the is...
 5.6.39: If p is a prime number and a is a positive integer not divisible by...
 5.6.40: Let m1, m2, , mn be pairwise relatively prime positive integers (th...
 5.6.41: The remaining step in the proof of the RSA algorithm is to show tha...
Solutions for Chapter 5.6: The Mighty Mod Function
Full solutions for Mathematical Structures for Computer Science  7th Edition
ISBN: 9781429215107
Solutions for Chapter 5.6: The Mighty Mod Function
Get Full SolutionsSince 41 problems in chapter 5.6: The Mighty Mod Function have been answered, more than 10884 students have viewed full stepbystep solutions from this chapter. Chapter 5.6: The Mighty Mod Function includes 41 full stepbystep solutions. Mathematical Structures for Computer Science was written by and is associated to the ISBN: 9781429215107. This textbook survival guide was created for the textbook: Mathematical Structures for Computer Science, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions.

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

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Column space C (A) =
space of all combinations of the columns of A.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Outer product uv T
= column times row = rank one matrix.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.