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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)[reply]

@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)[reply]

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

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

Are

n=-2,0

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]