Proof

We prove the first equality. The proof of the second is left to the reader.

We certainly have:

min{ord_p(a), ord_p(b)} ord_p(a)

and

min{ord_p(a), ord_p(b)} ord_p(b).

Hence the right-hand side of the equality is a common divisor of a and b. On the other hand, if ord_p(ggd(a,b)) = r > 0 for some prime p, then p divides both a and b so that we can conclude that r ord_p(a) en r ord_p(b). Hence

  _p prime p^{min{ord_p(a), ord_p(b)}}

equals gcd(a,b).