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Math Help - Prove Congruence

  1. #1
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    Prove Congruence

    I have to show that if p is a prime number, then the congruence x ≡ 1 (mod p) has only the solutions x ≡ 1 and x ≡ -1.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by page929 View Post
    I have to show that if p is a prime number, then the congruence x ≡ 1 (mod p) has only the solutions x ≡ 1 and x ≡ -1.
    Assume x^2 \equiv 1 \text{ mod }p. That is, p divides x^2-1. This is a difference of two squares, and so split it up and realise that p divides one of the two factors, because p is prime...
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