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Homework Help: Base b (not 10) proof (<=)

  1. Jun 29, 2010 #1
    1. The problem statement, all variables and given/known data

    Whenever base b is even (b=2k for some integer k) a number H=(d(n-1)d(n-2)...d1d0)b is even <=> d0 is even.

    2. Relevant equations

    3. The attempt at a solution
    I have formulated a proof for the forward direction (=>) but I am having trouble getting started on a proof for the backwards direction (<=) starting with the fact that d0 is even.
    Any advice would be much appreciated. I know all the definitions I'm just unsure how to build from d0 is even.
     
  2. jcsd
  3. Jun 29, 2010 #2

    Mark44

    Staff: Mentor

    You don't show what you've done, so this might or might not be useful.

    H=(d(n-1)d(n-2)...d1d0)b
    = d(n-1) * bn - 1 + d(n-2) * bn - 2 + ... + d1*b + d0

    You have d0 being even. What can you say about b raised to any positive power? Is it even or odd? What about multiples of even or odd numbers?
     
  4. Jun 29, 2010 #3
    even * even = even always
    odd * even = even always
    even + odd = odd

    I think my problem was identifying the problem correctly

    I rewrote it this way

    b = 2k, k is a natural number, H=(d(n-1)d(n-2)...d1d0)b is even <=> do is even

    shouldnt be too hard from there
     
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