A community for students.
Here's the question you clicked on:
 0 viewing
cwrw238
 4 years ago
Help  my grandson came up with this question:
Prove that every natural number is either even or odd.
cwrw238
 4 years ago
Help  my grandson came up with this question: Prove that every natural number is either even or odd.

This Question is Closed

cwrw238
 4 years ago
Best ResponseYou've already chosen the best response.0 his teacher suggested proof by induction

joemath314159
 4 years ago
Best ResponseYou've already chosen the best response.0i gave a solution to this problem a few semesters ago in some math class, but my professor didnt like it <.< im going to post it to see what others think. one sec let me see if i can find it.

joemath314159
 4 years ago
Best ResponseYou've already chosen the best response.0i saw what was wrong with my answer, never mind. I would just say that it is impossible to solve the equation:\[2n=2m+1\]with natural numbers n and m, so its impossible to have a natural number that is both even and odd. Not too sure how you would do that my induction, since a number being even or odd doesnt really depend on previous numbers being even or odd.

jhonyy9
 4 years ago
Best ResponseYou've already chosen the best response.1let n one number from set of natural numbers N ,so than odd even 2n+1 2n 1=2*0+1=1 2=2*1=2 3=2*1+1=3 4=2*2=4

Jemurray3
 4 years ago
Best ResponseYou've already chosen the best response.4Alrighty, lets give induction a go. First, we need to define even and odd. The standard definitions are as follows: An integer n is even if it can be expressed in the form n = 2k, where k is some integer An integer n is odd if it can be expressed in the form n = 2k + 1, where k is some integer We are now equipped to prove the statement via induction. We assume that the natural number n is either even or odd. We seek to show that this implies that the natural number (n+1) is also either even or odd. Case 1: n is an even number If n is an even number, then it can be expressed as n = 2k, with k some integer. Thus, n+1 = 2k+1, and by the definition of an odd number, n+1 is odd. Case 2: n is an odd number If n is an odd number, then it can be expressed as n = 2k + 1, with k some integer. Thus, n+1 = 2k+1 +1 = 2k+2 = 2(k+1). Since k is an integer, k+1 is an integer, and thus by the definition of an even number, n+1 is even. In either of these two cases, n+1 is either even or odd. Finally, we show that n= 1 is either even or odd. Since k=0 is an integer, note that 1 can be expressed as n = 2k+1 > 1 = 2(0) + 1. Thus n is odd, and more generally, n=1 is either even or odd. By induction, we may say that any natural number n is either even or odd.

cwrw238
 4 years ago
Best ResponseYou've already chosen the best response.0thnx guys  number theory is not my thing
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.