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Proving a sufficient condition, can someone check my work

  1. Sep 18, 2006 #1
    ello ello!

    I think i did this right but not sure! The directions are: Determine whether the statement is true or false. Prove the statement directly from the definitions or give a counter exmaple if it is false.

    A sufficient condition for an integer to be divisble by 8 is hat it be divisble by 16.

    [tex]\forall[/tex] integers n, if n is divisble by 8, then n is divisble by 16. This is a true statement.
    Proof: Suppose n is an integer divisble by 8. BY definition of divisbility, n = 8k for some integer k. But, 8k = 4*2k, and 2k is an integer becuase k is. Hence n = 4*(some integer) and so n is divisble by 16.

    Thanks!
     
  2. jcsd
  3. Sep 18, 2006 #2

    radou

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    What exactly do you mean by 'divisible'? Is it that the quotient must be an integer?
     
  4. Sep 18, 2006 #3

    StatusX

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    Where did this come from? You've shown n is divisible by 4, not 16.
     
  5. Sep 18, 2006 #4

    arildno

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    16n=8(2n)
    Maybe that could be used for something.
     
  6. Sep 18, 2006 #5

    shmoe

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    Take n=8. n is divisble by 8, but n is not divisible by 16.

    You tried to prove:

    "If n is divisible by 8 then it is divisible by 16".

    but in fact you showed:

    "if n is divisible by 8 then it is divisible by 4"


    But they actually claimed:

    "A sufficient condition for an integer to be divisble by 8 is hat it be divisble by 16."

    In otherwords, "if n is divisible by 16 then it is divisible by 8."
     
  7. Sep 18, 2006 #6
    Thanks guys!

    OKay i rewrote it using, ""if n is divisible by 16 then it is divisible by 8.""

    [tex]\forall[/tex] integers n, if n is divisble by 16, then n is divisble by 8. This is a true statement.

    Proof: Suppose n is an integer divisble by 16. By definition of divsibilty, n = 16k for some integer k. But, 16k = (8)(2k), and 2k is an integer because k is. Hence n = 8(some integer) and so n is divisble by 16.

    I think i had it switched around as shmoe pointed out
     
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