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.
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 expressible
The first integer not expressible is an odd number
=> K = 1 + 2^c + 2^2c + .... + 2^(x-1)c
The first odd number not expressible as above is the required
3 = 1 + 2^1
5 = 1 + 2^2
7 = 1 + 2^2 + 2^3
9 = 1 + 2^4
11 = 1 + 2 + 2^3 (not in GP)
So, 11 is the smallest integer
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Posted by Praneeth
on 2010-01-08 10:12:37 |
Solution
