Question

Help - my grandson came up with this question:
Prove that every natural number is either even or odd.

Answer 1

- his teacher suggested proof by induction

Answer 2

i 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.

Answer 3

i 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.

Answer 4

let 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

Answer 5

Alrighty, 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.

Answer 6

thnx guys - number theory is not my thing