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Given any two distinct odd primes $p$,$q$ can we find a Carmichael number divisible by $pq$?

If so, given any two distinct odd primes $p$,$q$ can we find a Carmichael number divisible by $pq$, wich have exactly 4 factors?

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    What's the question? Are you, possibly, asking "given any two distinct odd primes $p,q$ can we find a Carmichael number divisible by $pq$?" ? – 2017-01-06
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    Yes, I edited my question – 2017-01-06

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No.

Suppose $p\,|\,q-1$. If $n$ is a Carmichael number divisible by $q$ we must have $q-1\,|\,n-1$. Hence $p$ divides $n-1$, so $p$ can not divide $n$.

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    Short and clear, thanks – 2017-01-06
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    And what about, if gcd($p$,$q-1$) = 1 and gcd($q$,$p-1$) = 1 – 2017-01-06
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    I don't know. If you asked me to guess, I'd say that the obstruction I illustrated was the only one...but I am certainly not sure of that and I don't know how to go about constructing such $n$. So far as I am aware, it isn't even known that there are infinitely many Carmichael numbers divisible by a fixed prime (but, again, I could have that wrong). – 2017-01-06