Prove a randomly created message has probability less than $2^{-1023}$ of sharing a factor with n, where $n=pq$ is a product of 2 primes with 1024 binary digits. We can assume the likelihood of a message falling into each residue class is equally likely.
I first have started by proving that that the probability of an integer being relatively prime to n is $\frac{(p-1)(q-1)}{pq}$. Then, I can use this to prove that the probability of an integer not being relatively prime to n is $1-\frac{(p-1)(q-1)}{pq}=\frac1p+\frac1q-\frac1{pq}$. However, I'm not exactly sure where to go from here. If I can find p and q, or at least bounds for p and q, I belive I can figure out the inequality, but the fact that n has so many digits is where I'm stuck.
If $p$ has $1024$ binary digits, it is greater than $2^{1023}$ and the same for $q$. That gets a probability less than $2^{-1022}$. I don't know where to get the last factor $2$.