Let n be an odd positive integer. Consider a list of n consecutive integers, not
necessarily starting with 1. Show that the average is the middle number (that is the
number in the middle of the list when they are arranged in an increasing order). What
is the average when n is an even positive integer instead. We learned that for the odd numbers, we would have to show why n-1/2(2k+n)+(k+n) all over n equals k+(n+1)/2.
Let n be an odd positive integer. Consider a list of n consecutive integers, not necessarily...
3. Consecutive Sums a. (4 pts) Write 90 as the sum of consecutive positive integers in as many ways as possible. b. (4 pts) If a number can be written as n (d)(t) where d is an odd number of the form 2k + 1 and d is greater than 1, show symbolically how n can be written as the sum of consecutive numbers. Illustrate this with one example from part a. c. (4 pts) State a conjecture identifying the...
Show that every positive integer n, there is a string of n consecutive integers where first integer is even, the second is divisible by a perfect square(other than 1), the third by a perfect cube(other than 1), etc..., and the nth is divisible by the nth power of an integer(other than 1). Then find an example for n = 3.
15. For any integer k, which of the following represents three consecutive even integers? A 2k, 4k, 6k B k, k + 1, k + 2 C k, k + 2, k + 4 D 4k, 4k + 1, 4k + 2 E 2k, 2k + 2, 2k + 4
Write a Java program to prompt for inputting an integer N, then enter N integers (a loop is needed), print the numbers user entered, and the amount of even and odd numbers (zero is even number). (1) Prompt for the user to input an integer and output the integer. (1 pts) Enter an integer: You entered: 5 (2) Prompt for the user to input N integers, output the numbers entered and the amount of even and odd numbers (9 pts)...
please answer questions #7-13
7. Use a direct proof to show every odd integer is the difference of two squares. [Hint: Find the difference of squares ofk+1 and k where k is a positive integer. Prove or disprove that the products of two irrational numbers is irrational. Use proof by contraposition to show that ifx ty 22 where x and y are real numbers then x 21ory 21 8. 9. 10. Prove that if n is an integer and 3n...
Assume n is an integer. Prove that n is odd iff 3n2 + 4 is odd. Remember that to prove p iff q, you need to prove (i) p → q, and (ii) q → p. Use the fact that any odd n can be expressed as 2k + 1 and any even n can be expressed as 2k, where k is an integer. No other assumptions should be made.
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.
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...
Definition 1.6: An integer n is odd if n = 2k+1 for some integer k
EXERCISE 1.28. Show that for every positive integer k, there exist k consecutive composite integers. Thus, there are arbitrarily large gaps between primes. EXERCISE 1.12. Show that two integers are relatively prime if and only if there is no one prime that divides both of them.