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

= 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)
 * 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:
 * $$\frac{n}{n/2+1/2}=\frac{2n}{n+1}=2-\frac{2}{n+1}\in\Z$$
 * 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)