Determine the smallest positive integer n for which there exists a set
{s₁, s₂,..., s_n} consisting of n distinct positive integers such that:

(1 - 1/s₁) (1 - 1/s₂)....(1 - 1/s_n) = 51/2010

1. 51/2010 = 17/670 = (1/670)*17
2. Since 1/2*2/3*3/4*4/5...(n-1)/n=1/n, 1/670 can be expressed in 669, but no fewer, distinct terms of the set.      
3. Since 2/1*3/2*4/3*5/4...n/(n-1) = n, cancelling the first 17 terms of the set supplies the desired result, 17/670.      
4. But equally, we could ignore the first term, 2/1, starting with the second, 3/2, in which case we could cancel twice as many terms, or the third, 3/4, in which case we could cancel three times as many terms, and so on.      
5. 17 divides 670 39 times, with a remainder of 7. We keep the first 38 terms, eliminating (38*17) superfluous fractions, and the last 7, for a total of 45, which is the minimal such representation:     

1/2*2/3*3/4...38/39=1/39

     
1/39*663/664*664/665*665/666*666/667*667/668*668/669*669/670 = 17/670  

Nice problem.

 

Edited on February 3, 2014, 8:35 am

Posted by broll on 2014-02-03 07:00:39