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Factorials and how many zeroes are at the end

POSTED BY: Peter Burbery
4 Replies
Posted 1 year ago

Cool post Peter! Numbers with lots of trailing zeroes before the decimal point appear many unexpected places in discrete mathematics and combinatorics, such as the number of possible Rubik's Cube scrambles (43,252,003,274,489,856,000).

As you have shown, it tends to happen for numbers which can be derived as the product of several combinatorics functions (e.g. permutations and combinations) like the number of possible Rubik's cube scrambles, or a number which is a factorial of a very large integer.

POSTED BY: Chase Marangu

For arbitrary bases I think this should do what you want.

factorialZeros[n_Integer, b_Integer] /; n > 1 && b > 1 := 
 Module[{pfacs, pexpons},
  {pfacs, pexpons} = Transpose[FactorInteger[b]];
  Min[Floor[
    Map[Sum[Floor[n/#^j], {j, Floor[Log[#, n]]}] &, pfacs]/pexpons]]
  ]

For example:

In[86]:= factorialZeros[243, 12]

(* Out[86]= 118 *)
POSTED BY: Daniel Lichtblau

Would Block work here because it takes less time or should Module be used?

POSTED BY: Peter Burbery
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