Finding all the zeroes in $100!$

Question

Is there a way to find all the $0$s in $100!$? (Including zeroes that come between two non-zero numbers)

I know that to find the $0$s at the end we can use the greatest integer method. I was just curious if a method exists to find the remaining zeroes, too. (Of course without finding the actual value of $100!$ and counting, as is done in the only answer to this previous question: How many zeroes are in 100!.)

What John said. The other similar question did not get a proper answer. I'm not sure how this should be handled though. — DRF, Apr 19 '15 at 08:52
There was no need to post a new question, as this ask exactly the same as the old one! — Mariano Suárez-Álvarez, Apr 19 '15 at 09:49
What does finding all zeroes mean. Do you mean finding number of them and also positions? — Martin Sleziak, Feb 22 '16 at 14:07

score 1 · Answer 1 · answered Apr 19 '15 at 09:39

1

One way is to compute it!

In[2]:= Count[IntegerDigits[100!], 0]
Out[2]= 30

This computation took 0.000033 seconds in Mathematica.

answered Apr 19 '15 at 09:39

Mariano Suárez-Álvarez

135,076

(I seriously doubt there is a better way) – Mariano Suárez-Álvarez Apr 19 '15 at 09:39
The sequence $2, 30, 472, 5803, 68620, 782336, \dots$ is the number of zero digits in $10^k!$ starting from $k=1$. This is not in the OEIS (yet?). The next one, $10^7!$, has something like $1.5\times 10^8$ digits, and probably one needs to do something smarter than my trivial code above to compute the number of zeroes. – Mariano Suárez-Álvarez Apr 19 '15 at 09:43

Finding all the zeroes in $100!$

1 Answers1