(a)
To prove that if both a and b are even, then
(a)
Explanation of Solution
If
(b)
To prove that if a is odd and b is even, then
(b)
Explanation of Solution
If b is even and a is odd, then it can be written
(c)
To prove that if both a and b are odd, then
(c)
Explanation of Solution
if both a and b are odd, therefore
Then, there exists
(d)
To design an efficient binary
(d)
Explanation of Solution
if
if
return
else return
end if
else
return
else
return
end if
end if
Want to see more full solutions like this?
Chapter 31 Solutions
Introduction to Algorithms
- The following is an addition problem in binary notation. Each letter represents a unique binary digit. Which letter represents 1 and which represents 0? Design an algorithm for solving problems like this. YXX+XYX=XYYY I can solve this problem but i am having trouble with creating an algorithm to solve this.arrow_forward6 7 8 9 10 ======= 11 12 13 14 15 16 17 18 19 20 21 22 XO la b 1 m Periodicity U 50 33 97 D XI E F G H I J L M N Use linear congruential generator approach to find Xi, what is the periodicity? Rename cells C4, C5, C6 and C7 as X0, a, b and m, respectively. RUDIMENTS OF RANDOM NUMBER GENERATION (1 OF 2) LINEAR CONGRUENTIAL GENERATORS (BYLEHMER 1951 A sequence of integers defined by the following recursive formula X = (ax +b) MOD m, 1 = 0, 1.... which means divide (ax, + b) by m, let X... be the remainder of this division a multiplier b increment m the modulus x0 seed or initial point 0 Parrow_forwardFibonacci principle states that: If we let Xn be the nth integer of the sequence, then the next (n+1)th integer is equal to the sum of nth and (n-1)th integers. Xn+1 = Xn+Xn-1 Using the principle, derive some equations relative to the Fibonacci numbers.arrow_forward
- 7 Using contraposition, prove that if 3n+4 is even, then n is even, where n is an integer.arrow_forwardAn integer n is called k-perfect if u(n) =kn. Note that a perfect number is 2-perfect. Showthat 120 = 23 • 3 · 5 is 3-perfect.arrow_forwardUse the Extended Euclidean Algorithm find x and y such that gcd(798, 111) = 798x + 111yarrow_forward
- Computer Science Design a divide-and-conquer algorithm for finding the minimum andthe maximum element of n numbers using no more than 3n/2comparisons.arrow_forwardEuclid’s Algorithm states the following: gcd(a, b) - a and b are integers b ≠ 0 r is the remainder of integer division At each step, the remainder, r, decreases by at least 1 r must eventually become 0 Use Euclid’s Algorithm to calculate gcd(96, 128).arrow_forwardProvide a counter example for: The number n is an even integer if and only if 3n + 2 is an even integer.arrow_forward
- The Modular Operation r mod m = r denotes that r is the remainder of the division of r by m. For example, 27 mod 4 = 3. If two integers have the same remainder, then they are equivalent. For example, 27 = 55 mod 4. An integer r is called prime if the only two positive integers that evenly divide it are 1 and r. Using these definitions, rewrite each of the following theorems using quantifiers and pred- icates. Note that the theorems are not precisely stated. You are allowed to use only the predicate Prime(r) that is True if r is a prime, and False otherwise. No other predicates can be used. You can also use either | or the mod definition to indicate that a number is divisible by another. Consider all mumbers as positive integers greater than 0. a. Lagrange's four-square theorem: Every natural number can be expressed as a sum of four integer squares. b. Kaplansky's theorem on quadratic forms (partial): Ay prime number p equivalent to 1 mod 16 can be represented by both or neither of the…arrow_forwardProve using mathematical induction that 20 + 21 + ... + 2n = 2n+1 - 1 whenever n is a nonnegative integer.arrow_forwardTry to compute gcd or lcm ( gcd(9, 6) or lcm(9,6) ) by the following algorithm, support it with a programming codearrow_forward
- Database System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill EducationStarting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSONDigital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSON
- C How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSONDatabase Systems: Design, Implementation, & Manag...Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage LearningProgrammable Logic ControllersComputer ScienceISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill Education