All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

 Positive Integer ≠ (2P – 2Q)/(2R – 2S) (Posted on 2009-11-07)
Determine the smallest positive integer X, such that X is not expressible in the form (2P – 2Q)/(2R – 2S), where each of P, Q, R and S is a positive integer.

 No Solution Yet Submitted by K Sengupta No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
 Solution Comment 3 of 3 |
`Let K = (2^P - 2^Q)/(2^R - 2^S)=> K = 2^Q (2^(P-Q)-1)/ {2^S (2^(R-S)-1)}=> K = 2^(Q-S) (2^(P-Q)-1)/(2^(R-S)-1)=> K = 2^a (2^b - 1)/(2^c - 1)For this to be integer, b should be multiple of c=> K = 2^a (2^cx - 1)/(2^c - 1)If K is not expressible, then K/2^a is also not expressibleThe first integer not expressible is an odd number=> K = 1 + 2^c + 2^2c + .... + 2^(x-1)cThe first odd number not expressible as above is the required3 = 1 + 2^15 = 1 + 2^27 = 1 + 2^2 + 2^39 = 1 + 2^411 = 1 + 2 + 2^3 (not in GP)So, 11 is the smallest integer`

 Posted by Praneeth on 2010-01-08 10:12:37

 Search: Search body:
Forums (0)