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 2006-04-16, 06:35 #1 mfgoode Bronze Medalist     Jan 2004 Mumbai,India 40048 Posts How many zeros? How many zeros are there in ( 10,000 ! ) ? Mally
2006-04-16, 10:16   #2
xilman
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260348 Posts

Quote:
 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.

Code:
64325 zeros in radix 2
4166 zeros in radix 16
Paul

Last fiddled with by xilman on 2006-04-16 at 10:18 Reason: Trivial format change.

2006-04-16, 10:44   #3
drew

Jun 2005

2×191 Posts

Quote:
 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. Code: 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.

2006-04-16, 11:06   #4
axn

Jun 2003

23×233 Posts

Quote:
 Originally Posted by mfgoode How many zeros are there in ( 10,000 ! ) ? Mally
Four

2006-04-16, 15:55   #5
mfgoode
Bronze Medalist

Jan 2004
Mumbai,India

22·33·19 Posts
Number of zeros

Quote:
 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.

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

2006-04-16, 17:13   #6
xilman
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Quote:
 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

Last fiddled with by xilman on 2008-10-25 at 11:26

2006-04-16, 17:27   #7
xilman
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Quote:
 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

Last fiddled with by xilman on 2006-04-16 at 17:28

 2006-04-16, 18:57 #8 ewmayer ∂2ω=0     Sep 2002 Repรบblica de California 2×11×13×41 Posts 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. Last fiddled with by ewmayer on 2006-04-16 at 18:57
 2006-04-16, 19:09 #9 akruppa     "Nancy" Aug 2002 Alexandria 2,467 Posts 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: Code: echo '10000!' | gp -q | tr -d "[1-9\n]" | wc 0 1 5803 Alex Last fiddled with by akruppa on 2006-04-16 at 19:29
2006-04-16, 19:21   #10
Andi47

Oct 2004
Austria

2·17·73 Posts

Quote:
 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: Code: 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!

2006-04-16, 20:01   #11
drew

Jun 2005

2×191 Posts

Quote:
 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?

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