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Number Theory Division

  1. Jul 6, 2008 #1
    1. The problem statement, all variables and given/known data
    If a, b < c, and d are positive integers, prove the following inferences.

    1. a|b [itex] \wedge [/itex] c|d [itex]\rightarrow[/itex] ac|bd
    2. a|b <=> ac|bc


    2. Relevant equations



    3. The attempt at a solution

    1.

    a|b = x, then b = ax

    c|d = y, then d = cy

    bd = axcy

    thus ac|bd = ac|axcy, and ac|axcy = xy

    therefore ac|bd = xy if a|b = x and c|d = y

    2.

    c|ac = a and c|bc = b

    so c|(ac|bc) = a|b
     
  2. jcsd
  3. Jul 6, 2008 #2

    HallsofIvy

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    Yes, and these are the "relevant equations" above

    I think it is simpler and clearer to write "bd= (ac)(xy) so ac|bd".

    This makes no sense "c|ac" is the statement "c divides ac" and is not equal to anything. You mean to say ac/c= a and bc/b= b.

    You want to prove "a|b <=> ac|bc. That's and "if and only if" statement and must be proved both ways:

    1) if a|b then b= ax for some x. bc= axc ...

    2) if ac|bc then bc= acy for some y...
     
  4. Jul 6, 2008 #3
    I know, thats what I meant "c divides ac". a|b = b/a right? Doesn't it follow that c|ac = ac/c?
     
  5. Jul 6, 2008 #4

    matt grime

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    No, the collection of symbols a|b means "b is a multiple of a, or equivalently, a divides b without remainder". The symbols b/a represent a (rational in this case) number.
     
  6. Jul 10, 2008 #5
    For a|b <=> ac|bc

    a|b = x
    ac|bc = y

    b = ax
    bc = acy

    bc/c = acy/c => b = ay

    If b = ax and b = ay then x = y

    a|b = x <=> a|b = y therefore a|b <=> ac|bc

    Is this correct?

    <=> is equivalence, not <->.
     
  7. Jul 11, 2008 #6

    matt grime

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    Please stop using the symbol a|b to mean the same as b/a. They are different.
     
  8. Jul 11, 2008 #7
    Uh ok... what's wrong with what I did? I did c|ac = a, before and that apparently thats wrong so I used bc/c = b, this time. How else do I cancel out the c?

    Should it be c|bc = c|acy => b = ay ?

    Is my solution correct or wrong?
     
  9. Jul 11, 2008 #8
    A vertical bar means "divides." matt grime is just making a point about semantics and the use of symbols.

    It's not the same as the "fraction bar."
     
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