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If a positive integer multiple of $864$ is chosen randomly, with each multiple having the same probability of being chosen, what is the probability that it is divisible by $1944$?

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    If every one of the infinitely many multiples of$864$has the same probability $p$ of being chosen, then $p$ can only be zero. So your question doesn't make sense; you cannot randomly choose a multiple in this way.2012-03-25

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Although there is no uniform distribution on the integers, it's generally taken for granted that we work with a uniform distribution on, say, $\{-N,\dots,N\}$ and let $N\to\infty$. Now, maybe this will help:

$\rm ad|bd \iff a|b, \quad thus \quad a|bn\iff \frac{a}{\gcd(a,b)}\big\vert\left(\frac{b}{\gcd(a,b)}n\right)$

$\rm \gcd(x,y)=1 \implies(x|yz \iff x|z)$

This entails $1944|864n \iff 9|4n \iff 9|n $; what's the probability an integer is divisible by $9$?

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    It might be worth noting that in this case this probability is basically inherted from the natural uniform distribution on $Z/9Z$2012-03-25