Here is an explicit description of the gcd and lcm in terms of
prime factorizations.
Theorem
If a and b are positive integers, then
If a and b are positive integers, then
gcd(a,b) = 
p
prime pmin{ordp(a),ordp(b)} ,
lcm(a,b) = p prime
pmax{ordp(a),ordp(b)}.
In particular we have
a · b =
ggd(a,b)
· lcm(a,b).