Prove that the highest common factor of

$\displaystyle q^{n (n-1)/2}$ and $\displaystyle \prod_{1}^n (q^i -1)$ = 1, where $\displaystyle q = p^m$ for some prime p and natural number m.

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- Feb 23rd 2009, 03:12 PMAmanda1990Prove hcf(a,b) =1 where a,b are as follows...
Prove that the highest common factor of

$\displaystyle q^{n (n-1)/2}$ and $\displaystyle \prod_{1}^n (q^i -1)$ = 1, where $\displaystyle q = p^m$ for some prime p and natural number m. - Feb 25th 2009, 10:30 PMtah
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

$\displaystyle p | \prod_i(q^i-1) $

which implies, for some $\displaystyle i,\ p | (q^i-1)$ (property of prime elements) which is impossible. So the hcf is 1.