Consider the integers 715,798, 451, and 10. Using the pigeonhole principle, show that there exists an...
Consider the integers 429, 1330, 414, and 46. Using the pigeonhole principle, show that there exists an integer x such that ? =429?+414 and ?=1330?+46 for some integers p and q.
Pigeonhole Principle
2. (10 pts) Using the Pigeonhole Principle, show that in every set of 100 integers, there exist two whose difference is a multiple of 37.
Show your work, please
1. Counting and Pigeonhole Principle (a). A set of four different integers is chosen at random between 1 and 200 (inclusive). How many different outcomes are possible? (b). How many different integers between 1 and 200 (inclusive) must be chosen to be sure that at least 3 of them are even? (C). How many different integers between 1 and 200 (inclusive) mu be chosen to be sure that at least 2 of them add up to...
Please send me solutions for the above five
questions.
The questions are based on Pigeonhole Principle.
3. A shop contains twelve samples of read shirts, seven samples of white shirts, and N samples of blue shirts. Suppose that the smallest K such that choosing K samples from the collection guarantees that you have six samples of the same color of shirt is K-15. What is N? 4. Show that among any n1 positive integers not exceeding 2nthere must be integer...
Question 9: Let S be a set consisting of 19 two-digit integers. Thus, each element of S belongs to the set 10, 11,...,99) Use the Pigeonhole Principle to prove that this set S contains two distinct elements r and y, such that the sum of the two digits of r is equal to the sum of the two digits of y. Question 10: Let S be a set consisting of 9 people. Every person r in S has an age...
Use the well-ordering principle of natural numbers to show that for any positive rational number x ∈ Q, there exists a pair of integers a, b ∈ N such that x = a/b and the only common divisor of a and b is 1.
10. A natural number n is called attainable if there exists non-negative integers a and b such that n - 5a + 8b. Otherwise, n is called unattainable. Construct an 9 x 6 matrix whose rows are indexed by the integers between 0 and 8 and whose columns are indexed by the integers between 0 and 5 whose (x, y)-th entry equals 5x + 8y for any 0 < r < 8 and (a) Mark down all the attainable numbers...
(1 point) [4 Marks] Consider the following statements Q : There exists a real number n such that n? > 100 implies n < 10 and n > 0 R: If Tom is Ann's father then Jim is her uncle or Sue is her aunt or Mary is her cousin In English, what are the negations, converses, and contrapositives of Q and R? You do not need to justify the correctness of the statements. A. I am finished this question
Problem 2. In the Subset-Sum problem the input consists of a set of positive integers X = {x1, . . . , xn}, and some integer k. The answer is YES if and only if there exists some subset of X that sums to k. In the Bipartition problem the input consists of a set of positive integers Y = {y1, . . . , yn}. The answer is YES if and only if there exists some subset of X...
Prove that there exists infinitely many numbers of the form an = n(n+1)/2 , for some positive integer n, such that every pair an, am (for n != m) are relatively prime. [Hint: Assume there exists a finite sequence an1 < an2 < an3 < . . . < anm, where nj are increasing positive integers. Show that using those numbers we can construct a new number that fulfills the requirements.]