Wikipedia:Reference desk/Archives/Mathematics/2017 March 10

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

March 10

Odd number divisible by its half + 0.5

Is there any odd number n that can be divided by n/2 + 0.5? Is there any proof in wiki that this is impossible? Hofhof (talk) 20:25, 10 March 2017 (UTC)

@Hofhof: The only such odd number is 1. Let $n=2k+1$ for an integer k; then $n/2+1/2=k+1$, $n=2k+1\cong -1\mod (k+1)$ and -1 is not zero modulo (k + 1) unless n is 1.--Jasper Deng (talk) 20:40, 10 March 2017 (UTC)

If you allow negative numbers, -3 also works.--Jasper Deng (talk) 20:44, 10 March 2017 (UTC)

Or much simpler:

${\frac {n}{n/2+1/2}}={\frac {2n}{n+1}}=2-{\frac {2}{n+1}}\in \mathbb {Z}$

if and only if $n=-3,-2,0,1$. Ruslik_Zero 20:50, 10 March 2017 (UTC)

Are $n=-2,0$ odd?Hofhof (talk) 21:36, 10 March 2017 (UTC)

• Wouldn't it be more logical to designate 0 to be a trivial case and also require that any extension to negative numbers should likewise reverse the sign of the addend? Earl of Arundel (talk) 22:22, 10 March 2017 (UTC)