QUESTION 2 (Chapter 12, Exercises 787, p289) Let n be a positive integer. Let 1,1 E...
11. Let n be a positive integer and [r], [y e Zn. Show that the following conditions are equivalent. (i) []= [v] (ii) - y nr for some integer r. (ii) n/(x-y)
number thoery just need 2 answered 2. Let n be a positive integer. Denote the number of positive integers less than n and rela- tively prime to n by p(n). Let a, b be positive integers such that ged(a,n) god(b,n)-1 Consider the set s, = {(a), (ba), (ba), ) (see Prollern 1). Let s-A]. Show that slp(n). 1. Let a, b, c, and n be positive integers such that gcd(a, n) = gcd(b, n) = gcd(c, n) = 1 If...
1, and let σ be a permutation of {1, , n). Recall that for each integer m a) Let n 1, we denote ơm--σ ο . . . o σ. Show that n times b) Let 21, and let be a permutation of..,n consisting of a unique cycle of length n. Deduce from the previous question that there exists i e (1,..., n) such that i +c() )+22(n1). 1, and let σ be a permutation of {1, , n). Recall...
Question 1. Let E be the set of all positive integer multiples of 77. That is, | E = {" + N: 77%} (a) Prove that E is unbounded. (b) Consider the slightly modified set E' = {n€2: 7\n} Is E' bounded or unbounded? You do NOT need to prove your answer. Question 2. Show that the set v={*+1:n en} is unbounded.
Exercise 9. Let n 2 2 be a positive integer. Let a -(ri,...,^n) ER". For any a,y E R" sphere of radius 1 centered at the origin. Let x E Sn-be fixed. Let v be a random vector that is uniformly distributed in S"1. Prove: 10Vn
3 For each positive integer n, define E(n) 2+4++2n (a) Give a recursive definition for E(n). (b) Let P(n) be the statement E(n) nn1)." Complete the steps below to give a proof by induction that P(n) holds for every neZ+ i. Verify P(1) is true. (This is the base step.) ii. Let k be some positive integer. We assume P(k) is true. What exactly are we assuming is true? (This is the inductive hypothesis.) iii. What is the statement P(k...
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we would like to show that for all 0 SjSk-, the function satisfies the advancement operator equation (A -r)f0 (a) Show that this is true whenever J-0. You can use the fact that f(n) = crn satisfies (A-r)f = 0. (b) Suppose fm n) satisfies the equation when m s k-2 for every choice of c. Show that )...
1. Let m be a nonnegative integer, and n a positive integer. Using the division algorithm we can write m=qn+r, with 0 <r<n-1. As in class define (m,n) = {mc+ny: I,Y E Z} and S(..r) = {nu+ru: UV E Z}. Prove that (m,n) = S(n,r). (Remark: If we add to the definition of ged that gedan, 0) = god(0, n) = n, then this proves that ged(m, n) = ged(n,r). This result leads to a fast algorithm for computing ged(m,...
Ok = (6) Let n be a positive integer. For every integer k, define the 2 x 2 matrix cos(27k/n) - sin(2nk/n) sin(2tk/n) cos(27 k/n) (a) Prove that go = I, that ok + oe for 0 < k < l< n - 1, and that Ok = Okun for all integers k. (b) Let o = 01. Prove that ok ok for all integers k. (c) Prove that {1,0,0%,...,ON-1} is a finite abelian group of order n.
I got a C++ problem. Let n be a positive integer and let S(n) denote the number of divisors of n. For example, S(1)- 1, S(4)-3, S(6)-4 A positive integer p is called antiprime if S(n)くS(p) for all positive n 〈P. In other words, an antiprime is a number that has a larger number of divisors than any number smaller than itself. Given a positive integer b, your program should output the largest antiprime that is less than or equal...