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Am I making some mistake here:
Prob(p divides all three of K, L, M)=1/p^3
Prob(p does not divide all three) = 1 - 1/p^3
giving the final probability as the product of these terms over all p,
which is 1/zeta(3) = 0.8319...

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I don't understand the reason you choose to multiply those

particular factors (for each prime p).  The criterion is that

either 0 or 1 of K,L,M can be divisible by p, but no more than

1 of them.

 

--Dan Asimov