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statistics

Questions and Answers of Statistics

Determine which transposition errors the check digit of a UPC code finds.
Determine whether each of these 15-digit numbers is a valid airline ticket identification number. a) 101333341789013 b) 007862342770445 c) 113273438882531 d) 000122347322871
A parking lot has 31 visitor spaces, numbered from 0 to 30.Visitors are assigned parking spaces using the hashing function h(k) = k mod 31, where k is the number formed from the first three digits on
Can the accidental transposition of two consecutive digits in an airline ticket identification number be detected using the check digit?
Are each of these eight-digit codes possible ISSNs? That is, do they end with a correct check digit? a) 1059-1027 b) 0002-9890 c) 1530-8669 d) 1007-120X
Does the check digit of an ISSN detect every error where two consecutive digits are accidentally interchanged? Justify your answer with either a proof or a counterexample.
What sequence of pseudorandom numbers is generated using the linear congruential generator xn+1 = (3xn + 2) mod 13 with seed x0 = 1?
What sequence of pseudorandom numbers is generated using the pure multiplicative generator xn+1 = 3xn mod 11 with seed x0 = 2?
Find the first eight terms of the sequence of four-digit pseudorandom numbers generated by the middle square method starting with 2357.
Encrypt the message DO NOT PASS GO by translating the letters into numbers, applying the given encryption function, and then translating the numbers back into letters. a) f (p) = (p + 3) mod 26 (the
What is the decryption function for an affine cipher if the encryption function is c = (15p + 13) mod 26?
Suppose that the most common letter and the second most common letter in a long ciphertext produced by encrypting a plaintext using an affine cipher f (p) = (ap + b) mod 26 are Z and J, respectively.
Decrypt the message EABW EFRO ATMR ASIN which is the ciphertext produced by encrypting a plaintext message using the transposition cipher with blocks of four letters and the permutation σ of {1, 2,
The ciphertext OIKYWVHBX was produced by encrypting a plaintext message using the Vigenère cipher with key HOT. What is the plaintext message?
Suppose that when a long string of text is encrypted using a Vigenère cipher, the same string is found in the ciphertext starting at several different positions. Explain how this information can be
Show that we can easily factor n when we know that n is the product of two primes, p and q, and we know the value of (p − 1)(q − 1).
Encrypt the message UPLOAD using the RSA system with n = 53 · 61 and e = 17, translating each letter into integers and grouping together pairs of integers, as done in Example 8.
What is the original message encrypted using the RSA system with n = 43 · 59 and e = 13 if the encrypted message is 0667 1947 0671? (To decrypt, first find the decryption exponent d which is the
Describe the steps that Alice and Bob follow when they use the Diffie-Hellman key exchange protocol to generate a shared key. Assume that they use the prime p = 23 and take a = 5, which is a
Encrypt the message WATCH YOUR STEP by translating the letters into numbers, applying the given encryption function, and then translating the numbers back into letters. a) f (p) = (p + 14) mod 26 b)
Alice wants to send to all her friends, including Bob, the message "SELL EVERYTHING" so that he knows that she sent it. What should she send to her friends, assuming she signs the message using the
We describe a basic key exchange protocol using private key cryptography upon which more sophisticated protocols for key exchange are based. Encryption within the protocol is done using a private key
Decrypt these messages encrypted using the shift cipher f (p) = (p + 10) mod 26. a) CEBBOXNOB XYG b) LO WI PBSOXN c) DSWO PYB PEX
Suppose that when a string of English text is encrypted using a shift cipher f (p) = (p + k) mod 26, the resulting ciphertext is DY CVOOZ ZOBMRKXMO DY NBOKW. What was the original plaintext string?
Suppose that the ciphertext ERC WYJJMGMIRXPC EHZERGIH XIGLRSPSKC MW MRHMWXMRKYMWLEFPI JVSQ QEKMG was produced by encrypting a plaintext message using a shift cipher. What is the original plaintext?
a) What is the difference between a public key and a private key cryptosystem? b) Explain why using shift ciphers is a private key system. c) Explain why the RSA cryptosystem is a public key system.
a) Define what it means for a and b to be congruent modulo 7. b) Which pairs of the integers −11,−8,−7,−1, 0, 3, and 17 are congruent modulo 7? c) Show that if a and b are congruent modulo 7,
a) Define the greatest common divisor of two integers. b) Describe at least three different ways to find the greatest common divisor of two integers. When does each method work best? c) Find the
The odometer on a car goes to up 100,000 miles. The present owner of a car bought it when the odometer read 43,179 miles. He now wants to sell it; when you examine the car for possible purchase, you
Devise an algorithm for guessing a number between 1 and 2n − 1 by successively guessing each bit in its binary expansion.
Show that an integer is divisible by 9 if and only if the sum of its decimal digits is divisible by 9.
Prove there are infinitely many primes by showing that Qn = n! + 1 must have a prime factor greater than n whenever n is a positive integer.
Use Dirichlet's theorem, which states there are infinitely many primes in every arithmetic progression ak + b where gcd(a, b) = 1, to show that there are infinitely many primes that have a decimal
Show that every integer greater than 11 is the sum of two composite integers.
Show that Goldbach's conjecture, which states that every even integer greater than 2 is the sum of two primes, is equivalent to the statement that every integer greater than 5 is the sum of three
Prove that if f (x) is a nonconstant polynomial with integer coefficients, then there is an integer y such that f (y) is composite.
Use the Euclidean algorithm to find the greatest common divisor of 10,223 and 33,341.
Find gcd(2n + 1, 3n + 2), where n is a positive integer.
Adapt the proof that there are infinitely many primes (Theorem 3 in Section 4.3) to show that are infinitely many primes in the arithmetic progression 6k + 5, k = 1, 2, . . ..
Find four numbers congruent to 5 modulo 17.
Explain why you cannot directly adapt the proof that there are infinitely many primes (Theorem 3 in Section 4.3) to show that are infinitely many primes in the arithmetic progression 4k + 1, k = 1,
Determine whether the integers in each of these sets are mutually relatively prime. a) 8, 10, 12 b) 12, 15, 25 c) 15, 21, 28 d) 21, 24, 28, 32
For which positive integers n is n4 + 4n prime?
Find all solutions of the system of congruences x ≡ 4 (mod 6) and x ≡ 13 (mod 15).
Prove that 30 divides n9 − n for every nonnegative integer n.
Show that if p and q are distinct prime numbers, then pq−1 + qp−1 ≡ 1 (mod pq).
Show that the check digit of an ISBN-13 can always detect a single error.
Show that if d1d2 . . . d9 is a valid RTN, then d9 = 7(d1 + d4 + d7) + 3(d2 + d5 + d8) + 9(d3 + d6) mod 10. Furthermore, use this formula to find the check digit that follows the eight digits
The encrypted version of a message is LJMKG MGMXF QEXMW. If it was encrypted using the affine cipher f (p) = (7p + 10) mod 26, what was the original message?
Use the autokey cipher to encrypt the message THE DREAM OF REASON (ignoring spaces) using a) The keystream with seed X followed by letters of the plaintext. b) The keystream with seed X followed by
Show that if ac ≡ bc (mod m), where a, b, c, and m are integers with m > 2, and d = gcd(m, c), then a ≡ b (mod m/d).
Show that if n2 + 1 is a perfect square, where n is an integer, then n is even.
Develop a test for divisibility of a positive integer n by 8 based on the binary expansion of n.
Given n linear congruences modulo pairwise relatively prime moduli, find the simultaneous solution of these congruences modulo the product of these moduli.
Given a set of identification numbers, use a hash function to assign them to memory locations where there are k memory locations.
Given a message and a positive integer k less than 26, encrypt this message using the shift cipher with key k; and given a message encrypted using a shift cipher with key k, decrypt this message.
Given the positive integers a, b, and m with m > 1, find ab mod m.
Given a positive integer, find the Cantor expansion of this integer (see the preamble to Exercise 48 of Section 4.2).
Given a positive integer, find the prime factorization of this integer.
Given two positive integers, find their least common multiple.
There are infinitely many stations on a train route. Suppose that the train stops at the first station and suppose that if the train stops at a station, then it stops at the next station. Show that
Prove that 12 − 22 + 32 - · · ·+ (− 1)n−1n2 = (− 1)n−1 n(n + 1)/2 whenever n is a positive integer.
Prove that for every positive integer n, 1 · 2 + 2 · 3+· · ·+n(n + 1) = n(n + 1)(n + 2)/3.
Prove that Σnj =1 j4 = n(n+1) (2n+1) (3n2 +3n−1)/30 whenever n is a positive integer.
Let P(n) be the statement thatwhere n is an integer greater than 1. a) What is the statement P(2)? b) Show that P(2) is true, completing the basis step of the proof. c) What is the inductive
Prove that 2n > n2 if n is an integer greater than 4.
For which nonnegative integers n is 2n + 3 ≤ 2n? Prove your answer.
Prove that if h > −1, then 1 + nh ≤ (1 + h)n for all nonnegative integers n. This is called Bernoulli's inequality.
Prove that for every positive integer n,
Prove that H2n ≤ 1 + n whenever n is a nonnegative integer.
Let P(n) be the statement that 12 + 22 +· ∙ · ·+ n2 = n(n + 1) (2n + 1)/6 for the positive integer n. a) What is the statement P(1)? b) Show that P(1) is true, completing the basis step of the
Prove divisibility facts. Prove that 2 divides n2 + n whenever n is a positive integer.
Prove that a set with n elements has n(n − 1)/2 subsets containing exactly two elements whenever n is an integer greater than or equal to 2.
Devise a greedy algorithm that uses the minimum number of towers possible to provide cell service to d buildings located at positions x1, x2, . . . , xd from the start of the road.
What is wrong with this "proof" that all horses are the same color? Let P(n) be the proposition that all the horses in a set of n horses are the same color. Inductive Step: Assume that P(k) is true,
Prove that 12 + 32 + 52 +· · ·+ (2n + 1)2 = (n + 1) (2n + 1)(2n + 3)/3 whenever n is a nonnegative integer.
What is wrong with this "proof"? "Theorem" For every positive integer n, if x and y are positive integers with max(x, y) = n, then x = y. Basis Step: Suppose that n = 1. If max(x, y) = 1 and x and y
Use mathematical induction to prove that the derivative of f (x) = xn equals nxn−1 whenever n is a positive integer. (For the inductive step, use the product rule for derivatives.)
Suppose that m is a positive integer. Use mathematical induction to prove that if a and b are integers with a ≡ b (mod m), then ak ≡ bk (mod m) whenever k is a nonnegative integer.
Show that if A1, A2, . . . , An are sets where n ≥ 2, and for all pairs of integers i and j with 1 ≤ i < j ≤ n either Ai is a subset of Aj or Aj is a subset of Ai, then there is an integer i, 1
Prove that 3+3 · 5+3 · 52+ · · · + 3 · 5n = 3(5n + 1 − 1) / 4 whenever n is a nonnegative integer.
Let n be an even positive integer. Show that when n people stand in a yard at mutually distinct distances and each person throws a pie at their nearest neighbor, it is possible that everyone is hit
Construct a tiling using right triominoes of the 8 × 8 checkerboard with the square in the upper left corner removed.
Show that a three-dimensional 2n × 2n × 2n checkerboard with one 1 × 1 × 1 cube missing can be completely covered by 2 × 2 × 2 cubes with one 1 × 1 × 1 cube removed.
Show that a 5 × 5 checkerboard with a corner square removed can be tiled using right triominoes.
Use the principle of mathematical induction to show that P(n) is true for n = b, b + 1, b + 2, . . . , where b is an integer, if P(b) is true and the conditional statement P(k) → P(k + 1) is true
a) Find a formula for the sum of the first n even positive integers. b) Prove the formula that you conjectured in part (a).
Use strong induction to show that if you can run one mile or two miles, and if you can always run two more miles once you have run a specified number of miles, then you can run any number of miles.
Consider this variation of the game of Nim. The game begins with n matches. Two players take turns removing matches, one, two, or three at a time. The player removing the last match loses. Use strong
A jigsaw puzzle is put together by successively joining pieces that fit together into blocks. A move is made each time a piece is added to a block, or when two blocks are joined. Use strong induction
Prove that the first player has a winning strategy for the game of Chomp, introduced in Example 12 in Section 1.8, if the initial board is square.
Use strong induction to show that if a simple polygon with at least four sides is triangulated, then at least two of the triangles in the triangulation have two sides that border the exterior of the
In the proof of Lemma 1 we mentioned that many incorrect methods for finding a vertex p such that the line segment bp is an interior diagonal of P have been published. This exercise presents some of
Let E(n) be the statement that in a triangulation of a simple polygon with n sides, at least one of the triangles in the triangulation has two sides bordering the exterior of the polygon. a) Explain
Suppose that P(n) is a propositional function. Determine for which positive integers n the statement P(n) must be true, and justify your answer, if a) P(1) is true; for all positive integers n, if
Show that if the statement P(n) is true for infinitely many positive integers n and P(n + 1) → P(n) is true for all positive integers n, then P(n) is true for all positive integers n.
What is wrong with this "proof" by strong induction? "Theorem" For every nonnegative integer n, 5n = 0. Basis Step: 5 · 0 = 0. Inductive Step: Suppose that 5j = 0 for all nonnegative integers j with
Let P(n) be the statement that a postage of n cents can be formed using just 3-cent stamps and 5-cent stamps. The parts of this exercise outline a strong induction proof that P(n) is true for n ≥
Show that strong induction is a valid method of proof by showing that it follows from the well-ordering property.
Show that we can prove that P(n, k) is true for all pairs of positive integers n and k if we showa) P(1, 1) is true and P(n, k) → [P(n + 1, k) ∧ P(n, k + 1)] is true for all positive integers n

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