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Math Help - A proof in abstract algebra

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    A proof in abstract algebra

    Prove that if p is a prime and c is not congruent 0 (mod p), then cx is congruent b (mod p) has a unique solutin modulo p. That is, a solution exists, and any two solutions are congruent modulo p.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by rainyice View Post
    Prove that if p is a prime and c is not congruent 0 (mod p), then cx is congruent b (mod p) has a unique solutin modulo p. That is, a solution exists, and any two solutions are congruent modulo p.
    If there exists x, y such that cx \equiv cy \equiv b \text{ mod } p then this means that p|(cx-cy), by definition of congruence. So, p|c(x-y). As c \not\equiv 0 \text{ mod } p then p|(x-y). Thus, x \equiv y \text{ mod } p.
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  3. #3
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    Quote Originally Posted by Swlabr View Post
    If there exists x, y such that cx \equiv cy \equiv b \text{ mod } p then this means that p|(cx-cy), by definition of congruence. So, p|c(x-y). As c \not\equiv 0 \text{ mod } p then p|(x-y). Thus, x \equiv y \text{ mod } p.

    you are very helpful ~ thank you ^_^
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