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Math Help - Correct another congruence/mod proof, please!

  1. #1
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    Correct another congruence/mod proof, please!

    Can somebody please check my work on this proof? I would greatly appreciate it!

    Prove a ≡ b(mod m) if and only if there exists an integer k such that a = km+b.

    (1)
    If a ≡ b(mod m), then a = km+b, where k is an existing integer.
    Proof: (⇒)Suppose a ≡ b(mod m), then m|a-b.
    Thus a-b = km where k is an existing integer.
    Therefore, by addition, a = km+b.(⇐)

    (2)
    If a =km+b and k is an existing integer, then a ≡ b(mod m).
    Proof: Suppose a=km+b for an existing integer k.
    Then a-b=km. Thus m|a-b.
    Therefore a ≡ b(mod m).

    Hence, a ≡ b(mod m) ⇐⇒ a=km+b for an existing integer k.
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  2. #2
    MHF Contributor
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    Re: Correct another congruence/mod proof, please!

    looks fine to me
    Thanks from michaelgg13
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