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]

Let u = a + (b/c), where a, b, c are all integers and c>b>=0 and  b,c are relatively prime. 

Then [u] = a and {u} = b/c and Formula 1 (above) becomes
   a + (b/c) = a*b/c

Solving for a yields a = -b/(c-b). 
But the right hand expression is only integral if c - b = 1,
so c = b + 1 and a = -b

Substituting into the second formula,
5b/(b+1) + b/4 must be integral.

But this equals b(b+21)/4(b+1)

This can only be integral if b+1 divides b   (b = 0)
or if (b+1) divides (b+21).  

(b+1) divides (b + 21) only if c divides (c+20),
which means c must be (1,4,5,10, or 20)
and b therefore must be (0,3,4,9, or 19).

substituting,  only two of these values of b make b(b+21)/4(b+1) integral: 0 and 4.

Therefore, u = 0 or u = -4 + 4/5 = -16/5


    

Edited on March 9, 2008, 12:53 pm

Posted by Steve Herman on 2008-03-08 17:56:18