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Positive Integer ≠ (2P 2Q)/(2R 2S) (Posted on 2009-11-07) Difficulty: 4 of 5
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    
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Solution 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 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

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