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Modulus proof

  1. Jun 29, 2008 #1
    hi all,
    i was studying modulus when i came across this mod(xy) = modx mod y
    we consider 3 cases
    whereby both positive , both negative, or positive negative

    it reads here if we consider x and y to be negative
    it will be something like this

    mod xy = xy = - mod x ( - mod y ) = mod x mod y
    therefore proven

    i do not understand why it is -mod x and -mod y
    i know that it says that x and y has to be negative
    but isnt -mod x different from mod x ? eg if x is 2 , -mod x will be -2 , and mod x will be 2
    correct me if i'm wrong
    i'm lost
  2. jcsd
  3. Jun 29, 2008 #2

    matt grime

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    If x is negative, then mod(x)=-x. So what you've got there is that we've noted

    x = -(-x)
    y = -(-y)


    xy = (-(-x))(-(-y)) = (-mod x)(=mod y)

    if x,y <0.

    And, yes mod(x) and -mod(x) are obviously different, so you're correct there.
  4. Jun 30, 2008 #3
    hey matt ,
    but according to the proof
    it's mod xy = xy = - mod x ( - mod y ) = mod x mod y

    why can't it be
    mod xy = xy = mod x mod y
    why does it have to have an additional -mod x (-mod y) ?
  5. Jun 30, 2008 #4
    Because both x and y are negative and not equal to mod x or mod y respectively, but to -mod x and -mod y.
  6. Jun 30, 2008 #5

    matt grime

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    Those equalities many (and indeed do) hold, but you want to *prove* that they do. So you have to use the *definition* of modulus for -ve numbers, rather than just writing what you want to be true, i.e. go the extra mile and make things explicit.

    Think of it this way: you know that mod(x)=-x if x<0, now you want to show, using this knowledge, that if x,y<0, then mod(xy)=mod(x)mod(y).

    Since x and y are less than 0, xy>0, so mod(xy)=xy.

    Now, mod(x)=-x and mod(y)=-y, thus mod(x)mod(y)=(-x)(-y)=xy, so all is fine.

    All the proof does is put those things together in one line.
  7. Jun 30, 2008 #6


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    What definition of "modulus" are you using?. It seems to me that to prove something that fundamental you would want to use the precise definition.
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