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Subtract From Product, Get Integer (Posted on 2008-03-08) Difficulty: 3 of 5
Determine all possible rational u satisfying:

u = [u]*{u}, such that:
5*{u} - [u]/4 is an integer.

Note: [x] is the greatest integer ≤ x, and {x} = x - [x]

See The Solution Submitted by K Sengupta    
Rating: 2.0000 (2 votes)

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Solution computer search (spoiler; no proof) | Comment 1 of 3

If u is positive, [u]*{u} must be less than u, and so can't be equal, so there are no positive values of u that work.

Zero works trivially, and so is one of the values of u.

The following UBASIC program tests all negative rational u where the sum of the absolute values of the numerator and denominator is 20,000 or less:

   10   for T=1 to 20000
   20   for N=1 to T-1
   30    D=T-N
   35    if gcd(N,D)=1 then
   40     :U=-N//D
   50     :Iu=int(U)
   60     :Fu=U-Iu
   70     :if U=Iu*Fu then
   80       :if 5*Fu-Iu//4=int(5*Fu-Iu//4) then
   90         :print U
  100   next
  110   next

It finds only -16/5, where

-16/5 = -4 * 4/5  and

5 * 4/5 - (-4/4) = 4 + 1 = 5, an integer.

Now someone can find a proof that only 0 and -16/5 work.

 


  Posted by Charlie on 2008-03-08 12:50:42
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