Prove: Every natural number greater than 1 is of the form 2k or 2k+1 for some natural number k?

Question

please provide a written proof, like with words explaining how this is true... not a numerical proof. thanks!

Answer 1

You can easily see that every natural number is either odd or even. By definition, a even number is divisible by two, so it can be expressed as 2k where k is a natural number (positive integer, excluding 0).
An odd number can be expressed as 2k + 1.
Hence the result (follows from the abpve statements).

Answer 2

Whoops, I see that the theorem states natural number > 1.

So we again prove by induction the following theorem:

If n is a natural number, then n=1, or n is of the form 2k or 2k+1.

Proof by induction: n=1 is obviously true.

If n is 1, 2k, or 2k+1, then n+1 is 2, 2k+1, or 2k+2. In the first and third case, n+1 is of the form 2*1 and 2(k+1). In the middle case, n+1 is of the form 2k+1.

So, by induction, it is true for all natural n.

In particular, if n>1 is a natural number, then n!=1, so n must be of the form 2k or 2k+1 for some natural number k.

The answer from "7771313" (and Ravi, later) is typical circular reasoning. Odd numbers are defined as "not even," so it requires proof that odd numbers are of the form 2K+1. Basically, we know that all natural numbers are even or "not even," now you need to prove that the "not even" numbers are of the form "2k+1."

Amit, if you know that n is not even, how do you know that n-1 is even? That's basically assuming what needs to be proven, again.

Answer 3

Be n a natural number greater than 1

If n is even take k = n/2.

If n is odd, then it is greater than or equal to 3,
thus n - 1 >= 2
n - 1 is even, take k = (n - 1)/2
k is natural because it is an integer, and because
k = (n - 1)/2 >= 2/2 = 1

Answer 4

In type concept, a variety a is divisible by ability of a variety b if a / b is an entire type. considering 3/8 isn't an entire type we are saying that 3 isn't divisible by ability of 8. hence, n=3 _does_ artwork because of the fact 3^2 - a million = 8, and eight is divisible by ability of 8. What became probably mentioned at college is that n=2 would not artwork because of the fact 2^2 - a million= 3, and 3 isn't divisible by ability of 8. to that end n=2 is a counterexample.

Answer 5

that's trivial - every number is either even (2k) or odd (2k+1)