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 Power 2 The Cards (Posted on 2007-09-16)
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.

 Subject Author Date re: solution Charlie 2007-09-21 10:35:13 solution Charlie 2007-09-16 16:55:08

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