 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 19982 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.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

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

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

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

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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.

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).