Missing digit

Discussion in 'Math' started by Mark44, Jun 30, 2008.

  1. Mark44

    Thread Starter Well-Known Member

    Nov 26, 2007
    626
    1
    The number 2^{29} is interesting for the reason that it has nine digits in its base-10 form, and nine of the ten decimal digits are present.

    Without calculating 2^{29}, what's the missing digit?

    The problem is trivial if you calculate the number, using a calculator or computer or by longhand multiplication.

    Mark
     
  2. thingmaker3

    Retired Moderator

    May 16, 2005
    5,072
    6
    Without calculating? I can hear my father yelling at me now from 2000 miles away for making a guess instead of calculating!

    Okay. I always was a disobedient child.

    I'll guess "2." Somehow seems like the most "interesting" number to be missing.

    I'll calculate after posting, to know if my guess is correct or not.
     
  3. thingmaker3

    Retired Moderator

    May 16, 2005
    5,072
    6
    Wrong.

    That's what I get for guessing.
     
  4. Mark44

    Thread Starter Well-Known Member

    Nov 26, 2007
    626
    1
    Here's something that might serve as a hint:

    2^{6} = 64 \equiv 1 mod x
    2^{30} = (2^{6})^{5} \equiv (1^{5}) mod x = 1

    I'm doing modular arithmetic here, but I have not stated which modulo class I'm using. This is, after all, only a hint.
     
  5. Ratch

    New Member

    Mar 20, 2007
    1,068
    3
    To the Ineffable All,

    Taking the hint, and using the "casting out of 9's" http://mathforum.org/library/drmath/view/55831.html .

    2^29 = ((2^6)^4)*2^5=64*64*64*64*32

    Applying the casting out of 9's by multiplying the sum of the digits of the multiplicand and multipliers, and summing the product digits.

    (6+4)(6+4)(6+4)(6+4)(3+2)=50000==5

    Therefore the sum of the product digits of 2^29 must be 5.

    The only set of digits where this happens is:

    1+2+3+0+5+6+7+8+9 = 41 ==5

    Therefore, 4 is the missing digit of 2^29. Ratch
     
  6. Mark44

    Thread Starter Well-Known Member

    Nov 26, 2007
    626
    1
    Right you are!
     
  7. Dave

    Retired Moderator

    Nov 17, 2003
    6,960
    145
    Interestingly when I read this puzzle the first thing that jumped to my mind was the cast-out of 9s method - not because I had an idea of how this worked, but because of a puzzle jpanhalt posted a while back which employed this method. Something stuck in there from that.

    Good catch Ratch!

    Dave
     
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