Number theory during covid break -1

If two numbers are coprime, it means that they have no common prime factors. That means that for each $p$ , it cannot divide both numbers. The probability a number chosen at random is divisible by $p$ is just $\frac{1}{p}$ , so the probability that $p$ divides both is $p^{-2}$ and hence its complement is $1-p^{-2}$ . We take the product over all $p$ and this is just $\prod_p (1-p^{-2}) = \frac{6}{\pi^2}$ Similarly the general case is $\frac{1}{\zeta(n)}$

There is an interesting way to do the case of two numbers by considering that the probability something is coprime to $a$ is just $\frac{\phi(a)}{a}$ , i leave it as an interesting exercise to the reader to do that.