More elementary:  the probability that two positive integers have GCD=1 is
$6/\pi^2 = 1/\zeta(2)$ because the probability that a prime $p$ divides the GCD is
1/p^2 by considering each p by p block of pairs of positive integers.  More generally,
the probability that k positive integers have GCD 1 is $1/\zeta(k)$ by a similar argument.