TITLE: GCD of two large integers
QUESTION [5 upvotes]: For two random $d$ digit integers $a,b$, what is the probability 
$\gcd(a,b)<B$? Here $B$ is much much smaller than $a,b$.

REPLY [6 votes]: The probability that two integers, chosen uniformly at random from $1,2,\dots,n$, have greatest common divisor $g$, is (for large $n$) $\displaystyle{6\over\pi^2g^2}$. So the probability that the gcd is less than $B$ is $\displaystyle{6\over\pi^2}\sum_{g=1}^{B-1}{1\over g^2}$. 
A reference is J. Chidambaraswamy and R. Sitarmachandrarao, On the probability that the values of $m$ polynomials have a given g.c.d., Journal of Number Theory 26 (1987) 237–245. 
Information found at http://en.wikipedia.org/wiki/Greatest_common_divisor#Probabilities_and_expected_value
There is a good exposition at http://aperiodical.com/2013/06/cushing-your-luck-properties-of-randomly-chosen-numbers/
Choosing from $1,2,\dots,n$ isn't quite the same as choosing among $d$-digit numbers, but I suspect that for large $d$ the answers come out the same.