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Any large factorial will have a number of zero behind it, and one could write an expression to compute the number of trailing zeros, but how would one go about computing the non-zero end digits?

E.g. compute first 5 non-zero end digits of (10^12)!

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    $\begingroup$ Not a research level question, voting to close. $\endgroup$
    – Igor Rivin
    Nov 9, 2011 at 14:13
  • $\begingroup$ Indeed not research level, though it does lead to the $p$-adic Gamma function (here with $p=5$). $\endgroup$ Nov 9, 2011 at 14:18
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    $\begingroup$ Computing the first 5 non-zero end digits of (10^12)! is Project Euler problem 160. Clearly this kind of cheating shouldn't be encouraged. $\endgroup$
    – Guntram
    Nov 9, 2011 at 16:16

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This will be the least complete answer I've ever posted, but perhaps somebody else will be able to add to it. Not a complete answer (because I don't find any paper online), but I've heard Jean-Marc Deshouillers speak on his (with others, but I don't recall with whom) proof that the last non-zero base-$p$ digit of $n!$ forms a $q$-automatic sequence. I don't recall the hypothesis on $p$, nor the connection between $p$ and $q$.

But this answers (to a very limited extent) the real math-research level question hidden in this homework question. Or at least it would, if I could recall any details!

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  • $\begingroup$ What does "q-automatic" mean? $\endgroup$
    – Igor Rivin
    Nov 9, 2011 at 14:23
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    $\begingroup$ @Igor: there are many equivalent definitions, and the Allouche/Shallit book "Automatic Sequence" is beautifully written; I can recommend it easily for any "tourist" in the area. The wikipedia link is somewhat spartan: en.wikipedia.org/wiki/Automatic_sequence $\endgroup$ Nov 9, 2011 at 15:36

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