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Math Help - Prove hcf(a,b) =1 where a,b are as follows...

  1. #1
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    Prove hcf(a,b) =1 where a,b are as follows...

    Prove that the highest common factor of

    q^{n (n-1)/2} and \prod_{1}^n (q^i -1) = 1, where q = p^m for some prime p and natural number m.
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  2. #2
    tah
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    If the hcf you mentioned is the same as greatest common divisor (I don't what's the difference between them, if there is!!) then you can notice that any factor of q is power of p then p divide the hcf if it isn't equal to 1. Therefore
     p | \prod_i(q^i-1)
    which implies, for some i,\ p | (q^i-1) (property of prime elements) which is impossible. So the hcf is 1.
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