Question
Let $m=p q$ where $p$ and $q$ are distinct primes. Show that if $x$ is chosen at random in the range $1 \leq x \leq m$, then the probability that $\operatorname{gcd}(x, m) \neq 1$ is equal to $1 / p+1 / q-1 / m$.
Step 1
We need to find the probability that a randomly chosen number $x$ from the set $\{1, 2, \ldots, m\}$, where $m = pq$ (with $p$ and $q$ being distinct primes), is not coprime to $m$. This means $\gcd(x, m) \neq 1$. Show more…
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