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What does a=b (mod m) mean?

  1. Feb 28, 2005 #1
    What exactly does this mean?
    a=b (mod m)
    My teacher didn't attempt explain this but I'd still like to know.
     
  2. jcsd
  3. Feb 28, 2005 #2

    Galileo

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    [itex]a \equiv b \mod m[/itex], means [itex]a[/itex] is a multiple of [itex]m[/itex] plus [itex]b[/itex]. (a,b and m are integers)

    It means when you divide a by b, you get a rest of m.

    In algebra, a,b and m are not necessarily integers, but in most cases they are.
     
    Last edited: Feb 28, 2005
  4. Feb 28, 2005 #3
    So how do you solve (2/3)= x (mod 5) for x?
    I'm told that x=4, but if m<|b|, then that can't be right.
     
  5. Feb 28, 2005 #4

    xanthym

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    For integers "a", "b", and "m" (m > 0),
    a = b (mod m)
    if and only if (a - b) is exactly divisible by "m" (or equivalently, that "a" divided by "m" has the same remainder as "b" divided by "m").

    Example:
    10 = 4 (mod 3)
    ---> (10 - 4)/3 = Integer
    ---> {10 divided by 3} has same remainder as {4 divided by 3}


    ~~
     
    Last edited: Feb 28, 2005
  6. Feb 28, 2005 #5

    Galileo

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    I'm sorry, that remark was wrong. Just use the second one: :biggrin:

    [itex]a \equiv b \mod m[/itex] means division of a by b will give rest m.

    I'ven't seen fractions on the left side of congruences before, but they probably mean (multiplying both sides by 3):

    [tex]2 = 3x \mod 5[/tex]

    or

    [tex]2-3x = 0 \mod 5[/tex]
    So 2-3x must be a multiple of 5, so x =4 will work.
     
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