wu :: forums - Print Page


    
      
        wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

        riddles >> medium >> Improper Fraction Is An Integer
        
(Message started by: ThudanBlunder on Jul 22^nd, 2010, 5:47pm)

Title: Improper Fraction Is An Integer
Post by ThudanBlunder on Jul 22^nd, 2010, 5:47pm

For what positive integer values of n is the improper fraction g(n) = (12n³ - 5n² - 251n + 389)/(6n² - 37n + 45) also an integer?

Title: Re: Improper Fraction Is An Integer
Post by towr on Jul 23^rd, 2010, 1:45am

[hide]g(n) = 2n + 1007/(34(2n-9)) - 37/(17 ( 3n-5)) + 23/2 = 0 (mod 1)
1007/(34(2n-9)) - 37/(17 ( 3n-5)) = 1/2 (mod 1)
1007/(17(2n-9)) - 74/(17 ( 3n-5)) = 0 (mod 1)
1007/(2n-9) - 74/( 3n-5) = 0 (mod 17)

If both 1007/(2n-9) and 74/(3n-5) are integer, then n=14 is the only solution.
But while I think it's true that they can't be non-integer, I've yet to find a nice proof of it.[/hide]

Title: Re: Improper Fraction Is An Integer
Post by ThudanBlunder on Jul 27^th, 2010, 5:39am

[hide]g(n) can be rewritten as 2n + 9 + [(3n - 4)(5n + 4)/(3n - 5)(2n - 9)]

(3n - 4) and (3n - 5) being consecutive integers, they are relatively prime
and hence for g(n) be be an integer 3n - 5 must divide 5n + 4.

And when 3n - 5|5n + 4 it also divides 3(5n + 4) - 5(3n - 5) = 37

So 3n - 5 must divide at least one of the divisors of 37, which are 1, -1, 37, and -37.
This gives 2, 4/3, 14, and -32/3 as respective values of n, leaving 2 and 14 as possible answers.

g(2) = 13 - (2*14)/(1)*(-5) = 13 - 28/5 = 37/5 (not an integer)
and
g(14) = 37 + 4 = 41

So g(14) is the only solution, as towr suggested.
[/hide]