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

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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)

Template:Ping The only such odd number is 1. Let $n=2k+1$ for an integer k; then $n/2+1/2=k+1$, $n=2k+1\cong -1mod(k+1)$ 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)