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Discrete Math

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Let a, b, c, and d be integers, and let n be a positive integer. Prove that if a is congruent to c mod n and b is congruent to d mod n, then (a-b) is congruent to (c-d) mod n

  • Discrete Math -

    Given:
    a≡c mod n
    b≡d mod n
    Prove that (a-c)≡(b-d) mod n.

    Let
    a=kn+r ... k,r ∈ ℤ+
    c=ln+r ... l,r ∈ ℤ+
    Subtract:
    (a-c)=(k-l)n

    Similarly,
    (b-d)=(p-q)n ... p,q ∈ ℤ+

    Therefore
    (a-c)≡(b-d) mod n

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