How many zeros?

[mfgoode]

How many zeros are there in ( 10,000 ! ) ?
Mally

[xilman]

Originally Posted by mfgoode

How many zeros are there in ( 10,000 ! ) ?
Mally

On the assumption that 10,000 is written in decimal, the answers for radices 2 through 16 are given below. It was a completely trivial program to write, so computing the results for other radices and for the cases where 10,000 is not written in decimal is left as an addtional exercise.

64325 zeros in radix 2
28213 zeros in radix 3
18559 zeros in radix 4
12318 zeros in radix 5
11837 zeros in radix 6
7485 zeros in radix 7
7851 zeros in radix 8
6435 zeros in radix 9
5803 zeros in radix 10
3997 zeros in radix 11
7353 zeros in radix 12
3262 zeros in radix 13
3861 zeros in radix 14
4349 zeros in radix 15
4166 zeros in radix 16

Paul

[drew]

Originally Posted by xilman

On the assumption that 10,000 is written in decimal, the answers for radices 2 through 16 are given below. It was a completely trivial program to write, so computing the results for other radices and for the cases where 10,000 is not written in decimal is left as an addtional exercise.

64325 zeros in radix 2
28213 zeros in radix 3
18559 zeros in radix 4
12318 zeros in radix 5
11837 zeros in radix 6
7485 zeros in radix 7
7851 zeros in radix 8
6435 zeros in radix 9
5803 zeros in radix 10
3997 zeros in radix 11
7353 zeros in radix 12
3262 zeros in radix 13
3861 zeros in radix 14
4349 zeros in radix 15
4166 zeros in radix 16

Paul

Assuming base 10, and counting only trailing zeros, I get 2499.

[axn]

Originally Posted by mfgoode

How many zeros are there in ( 10,000 ! ) ?
Mally

Four

[mfgoode]

Originally Posted by xilman

On the assumption that 10,000 is written in decimal, the answers for radices 2 through 16 are given below. It was a completely trivial program to write, so computing the results for other radices and for the cases where 10,000 is not written in decimal is left as an addtional exercise.

5803 zeros in radix 10

Paul

Sorry to disappoint you Paul.
A fine effort but your answer on your 'trivial program' is way too high, and simply put, completely wrong.

I am not into programming, and all my puzzles or problems are by paper and pencil.

Since it is wrong for radix 10, I presume it is wrong for the rest of the radices you have left me as an excercise.

However it could be misleading to the others following this thread

Anyway, Happy Easter to you, and loved ones.
Mally

[xilman]

Originally Posted by mfgoode

Sorry to disappoint you Paul.
A fine effort but your answer on your 'trivial program' is way too high, and simply put, completely wrong.

I am not into programming, and all my puzzles or problems are by paper and pencil.

Since it is wrong for radix 10, I presume it is wrong for the rest of the radices you have left me as an excercise.

However it could be misleading to the others following this thread

Anyway, Happy Easter to you, and loved ones.
Mally

Sigh.

Assuming the simultaneously clever and fatuous answer of 4 is not correct, I invite you to count the zeros in the decimal expansion of 10,000! given in the attached output from pari/gp

You will find that there are indeed 5803 zeros in 10,000!, of which 2502 are contiguous at the end of the number. The decimal expansion begins 28462596809170545189, which shows the first two occurrences of a zero.


Paul

[xilman]

Originally Posted by mfgoode

Sorry to disappoint you Paul.
A fine effort but your answer on your 'trivial program' is way too high, and simply put, completely wrong.

Ok, since we disagree on such a simple process as counting, I suggest that you post your answer and your reasoning. I have already given mine in my previous post.


Paul

[ewmayer]

My answer for base-10 (5803 zeros) agrees with Paul's - that took all of roughly 15 seconds using the PARI-GP calculator and a simple text editor to verify. I don't know what the rest of y'all are smoking.

[akruppa]

Maybe it's some sort of trick question and Mally wants the number of zero '0' characters in the string "( 10,000! )", not the number of zero digits in the decimal expansion of factorial(10000).

For the latter, my count agrees with Paul's:

echo '10000!' | gp -q | tr -d "[1-9\n]" | wc
      0       1    5803

Alex

[Andi47]

Originally Posted by akruppa

Maybe it's some sort of trick question and Mally wants the number of zero '0' characters in the string "( 10,000! )", not the number of zero digits in the decimal expansion of factorial(10000).

For the latter, my count agrees with Pauls:

echo '10000!' | gp -q | tr -d "[1-9\n]" | wc
      0       1    5803

Alex

My count agrees with 5803 - using good ol' Derive for MS-DOS (0,2 seconds for the calculation of 10000!) and a text editor for counting.

If you take the "," for a decimal point, then you get 2 zeros in 10!

[drew]

Originally Posted by xilman

of which 2502 are contiguous at the end of the number.

Aside from running that program, which I haven't done, how can you rationalize that result? I solved it using the following, assuming the trailing zeros would correspond to the combined number of 5s among the factors of all of the numbers multiplied (2s are far more abundant):

floor(10,000/2^5)+10,000/2^4+10,000/2^3+10,000/5^2+10,000/5

The result I get is 2499, which differs from your answer by 3. Where did those other 3 zeros come from?