Question 1. Let E be the set of all positive integer multiples of 77. That is,...
Prove: Let k be a positive integer, and set n :=2k-1(2k – 1). Then (2k+1 – 1)2 = 8n +1 Prove: Let n be a positive integer, and let s and t be integers. Show that Hire (st) = n(s) in (t) mod n.
For an integer n > 0, consider the positive integer F. = 22 +1. (a) Use induction to prove that F. ends in digit 7 whenever n 2 is an integer (b) Use induction to prove that F= 2 + IT- Fholds for all neN. (c) Use (b) to prove that ged(F, F.) = 1 holds for all distinct nonnegative integers m, na (d) Use (e) to give a quick proof that there must be infinitely many primes! That is...
=(V, En) 5. Let n1 be an integer and define the graph Gn as follows {0,1}", the set of all binary strings of length n. Vn = Two vertices x and y are connected by an edge emu if and only if x and y differs in exactly one position. (a) (4 points) Draw the graph Gn for n = 1,2,3 (b) (4 points) For a general n 2 1, find |Vn and |En (c) (10 points) Prove that for...
1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers na +3n+6 and n2 + 2n +7 cannot be prime at the same time. (b) Find 15261527863698656776712345678%5 without using a calculator. (c) Let a be an integer number. Suppose a%2 = 1. Find all possible values of (4a +1)%6. 2. (Integers: %, =) (a) Suppose a, b, n are integer numbers and n > 0. Prove that (a+b)%n = (a%n +B%n)%n. (b) Let a,...
QUESTION 2 (Chapter 12, Exercises 787, p289) Let n be a positive integer. Let 1,1 E MEXa(R) satisfy the condition that Σ-laul 1 i < n. Show that Ic] < 1 for all c E spec(A). 1 for all
Let (xn) be a bounded sequence
of real numbers, and put u = lim supn→∞ xn . Let E be the set
consisting of the limits of all convergent subsequences of (xn).
Show that u ∈ E and that u = sup(E).
Formulate and prove a similar result for lim infn→∞ xn .
Thank you!
7. Let (Fm) be a bounded sequence of real numbers, and put u-lim supn→oorn . Let E be the set consisting of the limits of...
4. Let n be a positive integer. Z" is the set of all lists of length n whose entries are in Z. Prove that Z" is countable. (Hint: Find a bijection between Z"-1x Z and Z" and then use induction.)
4. Let n be a positive integer. Z" is the set of all lists of length n whose entries are in Z. Prove that Z" is countable. (Hint: Find a bijection between Z"-1x Z and Z" and then use induction.)
Let U be the set of all integers. Consider the following sets: S is the set of all even integers; T is the set of integers obtained by tripling any one integer and adding 1; V is the set of integers that are multiples of 2 and 3. a) Use set builder notation to describe S, T and V symbolically. b) Compute s n T, s n V and T V. Describe these sets using set builder notation
4) Let D be the set of all finite subsets of positive integers. Define a function (:2 - D as follows: For each positive integer n, f(n) =the set of positive divisors of n. Find the following f (1), f(17) and f(18). Is f one-to-one? Prove or give a counterexample.
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...