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Math Help - Primes and divisibility

  1. #1
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    Primes and divisibility

    Let p and q be different primes. Prove that pg | p^{q-1} + q^{p-1} - 1.

    Thx in advance!
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  2. #2
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    Try using Fermat's Little Theorem, which says that a^{p-1} \equiv 1 \ (mod \ p) for some prime p and integer a with (a,p)=1

    Also, a \equiv b \ (mod \ c) \Leftrightarrow c \ | \ a-b

    Also, a|a^x with x \geq 1

    Last thing you'll need is that if p|a and q|a then pq|a

    Hope that helps ... if you still can't see how I did it, just ask for more details
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  3. #3
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    Smile

    Quote Originally Posted by Bingk View Post
    Try using Fermat's Little Theorem, which says that a^{p-1} \equiv 1 \ (mod \ p) for some prime p and integer a with (a,p)=1

    Also, a \equiv b \ (mod \ c) \Leftrightarrow c \ | \ a-b

    Also, a|a^x with x \geq 1

    Last thing you'll need is that if p|a and q|a then pq|a

    Hope that helps ... if you still can't see how I did it, just ask for more details
    Thx, that was easy ...
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  4. #4
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    Quote Originally Posted by strelc View Post
    Let p and q be different primes. Prove that pg | p^{q-1} + q^{p-1} - 1.

    Thx in advance!
    Generalize this. Let n,m be positive integers with (n,m)=1 prove that nm | n^{\phi(m)}+m^{\phi(n)} - 1.
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