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If p is a prime number and a>1 is a natural number, then show that the greatest common divisor of the two number a-1 and (a^p - 1)/a-1 is either 1 or p
Let d=gcd{(a-1),(a^{p}-1/(a-1))}
then d=1 where both are prime to each other
or, d|{-(a-1)+(a-1)(a^{p}-1/(a-1))}
or,d|{a^{p}-1-(a-1)}
or, d|pq where q is some integer by fermat's theorem
or, d=p.
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