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 Power Division II (Posted on 2011-11-05)
Determine the highest power of 11 (base ten) that evenly divides N (base ten), where:

N = 12....221 (the digit one followed by 2001 twos followed by one)

*** For an extra challenge, solve this puzzle without using a computer program.

 No Solution Yet Submitted by K Sengupta Rating: 5.0000 (1 votes)

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 Analytic Solution Comment 2 of 2 |
Divide N by 11.  That is pretty easy to do by hand.  The result is the 2002 digit repunit 111......111.

This number has an even number of 1s, so it is divisible by 11.  Divide and get 101010...0101, a number alternating 1001 1s and 1000 0s.

The 21 digit number 101010101010101010101 equals 11*9182736455463728191.  Then 101010...0101 equals 11*9182736455463728191*(1 + 10^22 + 10^44 + ... + 10^181)

Factoring out the 11 leaves 9182736455463728191*(1 + 10^22 + 10^44 + ... + 10^181).  9182736455463728191 and 10^(2*n) are both 1 mod 11, then the whole number equals 182 mod 11 = 6 mod 11.  The no more 11s can be divided out.

Then the highest power of 11 which divides 122...221 is 11^3.

 Posted by Brian Smith on 2017-02-28 09:18:00

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