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Math Help - Congruence Proof

  1. #1
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    Congruence Proof

    Prove that if m and n are positive integers and if ab ≡ 1 (mod m) and
    a^n ≡ 1 (mod m), then b^n ≡ 1 (mod m).

    Any help would be greatly appreciated! I am lost with this one but I probably am just missing a very simple step. Thanks!
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  2. #2
    Senior Member Dinkydoe's Avatar
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    Given is ab\equiv 1 mod m. This implies that (ab)^n\equiv 1 mod m.

    Thus if a^n\equiv 1 mod n what does this say about (ab) ^n\equiv b^n mod m?
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by FatherMike View Post
    Prove that if m and n are positive integers and if ab ≡ 1 (mod m) and
    a^n ≡ 1 (mod m), then b^n ≡ 1 (mod m).

    Any help would be greatly appreciated! I am lost with this one but I probably am just missing a very simple step. Thanks!
    We have that 1\equiv a^n\equiv \left(ab\right)^n=a^nb^n\equiv 1\cdot b^n=b^n\text{ mod }m
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