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 No direct evaluation (Posted on 2015-06-09)
Given that 2^29 is a nine-digit number all of whose digits are distinct, determine which of the ten digits is missing.

Source: SMO contest

 No Solution Yet Submitted by Ady TZIDON No Rating

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 re: Solution | Comment 2 of 4 |
(In reply to Solution by Jer)

your answer is right but I must disagree with your reasoning as stated.  It needs to be clarified that this approach only works if the number has a non-zero digital root.

For example, take the 9-digit number 102345678.  It has 9 distinct digits which sum to 36 which is a multiple of 9 thus
102345678 = 0 mod 9
however the missing digit is not zero

so in general
if N is a 9-digit number consisting of 9 distinct digits, then
N= r mod 9
means that r is the missing digit UNLESS r=0 in which case the missing digit could be either 0 or 9

 Posted by Daniel on 2015-06-09 09:43:07

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