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Power 2 The Cards (Posted on 2007-09-16) Difficulty: 3 of 5
88,889 cards are consecutively numbered from 11,111 to 99,999.

These cards are now arranged in a line in any arbitrary order.

Can the 444,445 digit number formed in this manner be a power of 2?

  Submitted by K Sengupta    
Rating: 2.0000 (1 votes)
Solution: (Hide)
No. The 444,445 digit number formed in this manner be can never be a power of 2.

EXPLANATION:

If possible, let the number correspond to the form 2m, where m is an integer.

Then, it follows that:

10444,444 <= 2m < 10444,445
Or, 444,444*(log210) <= m < 444,445*log2(10)
Or, 1476411.0102 <= m < 1476414.3321
Or, m = 1476412, 1476413 or 1476414; giving:
m (Mod 6) = 4, 5 or 0
Or, 2^m(Mod 9) = 7, 5 or 1. .........(*)

Now, 111111 + 111112 + + 999999
= 999,999*500,000 - (5555*11111), which is congruent to 8(Mod 9).

Therefore, it follows that the 444,445 digit number must be congruent to 8(Mod 9). This contradicts (*) since 2^m cannot correspond to 8(Mod 9).

Consequently, the 444,445 digit number formed in this manner be can never be a power of 2.

-----------------------------------------------------

An alternative analysis and methodology has been posted by Charlie here and here.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Solutionre: solutionCharlie2007-09-21 10:35:13
SolutionsolutionCharlie2007-09-16 16:55:08
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