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)[reply]

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

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

Or much simpler:

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

if and only if ${\displaystyle n=-3,-2,0,1}$. Ruslik_Zero 20:50, 10 March 2017 (UTC)[reply]

Are ${\displaystyle n=-2,0}$odd?Hofhof (talk) 21:36, 10 March 2017 (UTC)[reply]

• 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)[reply]