Number Theory

[Prakash Kapadia]

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

[Shirsendu Roy]

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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