**A**
Exactly 40% !

This number arises as zeta(4)/zeta(2)^{2}.
For each *n*=1,2,3,..., the probability that
*a*,*b*)=*n**n*^{-2}/zeta(2), so the probability that
*a*,*b*)=gcd(*c*,*d*)=*n**n*^{-4}/zeta(2)^{2}*n*.

Can the answer 2/5 be proved by elementary means?

This can be regarded as a form of a question attributed to Wagstaff in problem B48 of R.K.Guy'sIs there any value of a xillion for which the probability is actually less than 40%?Unsolved Problems in Number Theory(Springer, 1981): “Wagstaff asked for an elementary proof (e.g., without using properties of the Riemann zeta-function) that [the product of(p over all primes equals 5/2].”^{2}+1)/(p^{2}-1)