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Math Help - Congruence help

  1. #1
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    Congruence help

    Let [a] = {x e Z | x and a are congruent mod m}

    This is the exercise:

    Describe the set [5] if m = 1. Show that [5] = [-1] in this case.

    any IDeas?
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  2. #2
    Junior Member
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    Quote Originally Posted by jzellt View Post
    Let [a] = {x e Z | x and a are congruent mod m}
    Describe the set [5] if m = 1. Show that [5] = [-1] in this case.
    To clarify did you mean,

    [a] = \{x \in \mathbb{Z} | x \equiv a \pmod m\}

    What part are you confused about?

    x and a are congruent mod m means the remainder of \frac{x}{m} is the same as the remainder of \frac{a}{m}

    Because 5 mod 1 = 0, because \frac{5}{1} = 5 (with no remainder) we have,
    [5] = {all integers that have a zero remainder when divided by 1} = {all integers} = \mathbb{Z}

    Show that [-1] is the set of all integers too.
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