1000!
Moderators: gmalivuk, Moderators General, Prelates

 Posts: 35
 Joined: Tue Oct 16, 2007 9:49 pm UTC
1000!
I'm going to assume many of you know this, but I still find it fun to ask.
How many zeros are at the end of 1000! (1000 factorial)
How many zeros are at the end of 1000! (1000 factorial)
Re: 1000!
none, but there's 3 in the middle
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Re: 1000!
The number of ending zeros will be the same as the number of times 5 divides 1000! (There will be more than enough 2s to pair up with the 5s to make 10s)
Thinking of 1000! as a product of 1000 factors...
5 divides 200 of the factors,
5^{2} divides 40 of them,
5^{3} divides 8 of them,
5^{4} divides 1 of them,
for a total of 249.
Thinking of 1000! as a product of 1000 factors...
5 divides 200 of the factors,
5^{2} divides 40 of them,
5^{3} divides 8 of them,
5^{4} divides 1 of them,
for a total of 249.
 LoopQuantumGravity
 Posts: 416
 Joined: Mon Oct 01, 2007 4:19 am UTC
Re: 1000!
I don't know, let's count!
4023872600770937735437024339230039857193748642107146325437999104299385\
1239862902059204420848696940480047998861019719605863166687299480855890\
1323829669944590997424504087073759918823627727188732519779505950995276\
1208749754624970436014182780946464962910563938874378864873371191810458\
2578364784997701247663288983595573543251318532395846307555740911426241\
7474349347553428646576611667797396668820291207379143853719588249808126\
8678383745597317461360853795345242215865932019280908782973084313928444\
0328123155861103697680135730421616874760967587134831202547858932076716\
9132448426236131412508780208000261683151027341827977704784635868170164\
3650241536913982812648102130927612448963599287051149649754199093422215\
6683257208082133318611681155361583654698404670897560290095053761647584\
7728421889679646244945160765353408198901385442487984959953319101723355\
5566021394503997362807501378376153071277619268490343526252000158885351\
4733161170210396817592151090778801939317811419454525722386554146106289\
2187960223838971476088506276862967146674697562911234082439208160153780\
8898939645182632436716167621791689097799119037540312746222899880051954\
4441428201218736174599264295658174662830295557029902432415318161721046\
5832036786906117260158783520751516284225540265170483304226143974286933\
0616908979684825901254583271682264580665267699586526822728070757813918\
5817888965220816434834482599326604336766017699961283186078838615027946\
5955131156552036093988180612138558600301435694527224206344631797460594\
6825731037900840244324384656572450144028218852524709351906209290231364\
9327349756551395872055965422874977401141334696271542284586237738753823\
0483865688976461927383814900140767310446640259899490222221765904339901\
8860185665264850617997023561938970178600408118897299183110211712298459\
0164192106888438712185564612496079872290851929681937238864261483965738\
2291123125024186649353143970137428531926649875337218940694281434118520\
1580141233448280150513996942901534830776445690990731524332782882698646\
0278986432113908350621709500259738986355427719674282224875758676575234\
4220207573630569498825087968928162753848863396909959826280956121450994\
8717012445164612603790293091208890869420285106401821543994571568059418\
7274899809425474217358240106367740459574178516082923013535808184009699\
6372524230560855903700624271243416909004153690105933983835777939410970\
0277534720000000000000000000000000000000000000000000000000000000000000\
0000000000000000000000000000000000000000000000000000000000000000000000\
0000000000000000000000000000000000000000000000000000000000000000000000\
000000000000000000000000000000000000000000000000
That number is much smaller than I'd have thought...
4023872600770937735437024339230039857193748642107146325437999104299385\
1239862902059204420848696940480047998861019719605863166687299480855890\
1323829669944590997424504087073759918823627727188732519779505950995276\
1208749754624970436014182780946464962910563938874378864873371191810458\
2578364784997701247663288983595573543251318532395846307555740911426241\
7474349347553428646576611667797396668820291207379143853719588249808126\
8678383745597317461360853795345242215865932019280908782973084313928444\
0328123155861103697680135730421616874760967587134831202547858932076716\
9132448426236131412508780208000261683151027341827977704784635868170164\
3650241536913982812648102130927612448963599287051149649754199093422215\
6683257208082133318611681155361583654698404670897560290095053761647584\
7728421889679646244945160765353408198901385442487984959953319101723355\
5566021394503997362807501378376153071277619268490343526252000158885351\
4733161170210396817592151090778801939317811419454525722386554146106289\
2187960223838971476088506276862967146674697562911234082439208160153780\
8898939645182632436716167621791689097799119037540312746222899880051954\
4441428201218736174599264295658174662830295557029902432415318161721046\
5832036786906117260158783520751516284225540265170483304226143974286933\
0616908979684825901254583271682264580665267699586526822728070757813918\
5817888965220816434834482599326604336766017699961283186078838615027946\
5955131156552036093988180612138558600301435694527224206344631797460594\
6825731037900840244324384656572450144028218852524709351906209290231364\
9327349756551395872055965422874977401141334696271542284586237738753823\
0483865688976461927383814900140767310446640259899490222221765904339901\
8860185665264850617997023561938970178600408118897299183110211712298459\
0164192106888438712185564612496079872290851929681937238864261483965738\
2291123125024186649353143970137428531926649875337218940694281434118520\
1580141233448280150513996942901534830776445690990731524332782882698646\
0278986432113908350621709500259738986355427719674282224875758676575234\
4220207573630569498825087968928162753848863396909959826280956121450994\
8717012445164612603790293091208890869420285106401821543994571568059418\
7274899809425474217358240106367740459574178516082923013535808184009699\
6372524230560855903700624271243416909004153690105933983835777939410970\
0277534720000000000000000000000000000000000000000000000000000000000000\
0000000000000000000000000000000000000000000000000000000000000000000000\
0000000000000000000000000000000000000000000000000000000000000000000000\
000000000000000000000000000000000000000000000000
That number is much smaller than I'd have thought...
I study theoretical physics & strings, and am a recipient of the prestigious Jayne Cobb Hero of Canton award.
And the science gets done and you make a neat gun
For the people who are still alive!
And the science gets done and you make a neat gun
For the people who are still alive!
Re: 1000!
Numerology time!
1000 in base 5 is 13000. The sum of the digits is 4. Subtract that from the original number and get 12441. Divide by 4, and we have 1444, which, in base 10, is 249, which is the answer we want.
1000 in base 5 is 13000. The sum of the digits is 4. Subtract that from the original number and get 12441. Divide by 4, and we have 1444, which, in base 10, is 249, which is the answer we want.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

 Posts: 35
 Joined: Tue Oct 16, 2007 9:49 pm UTC
Re: 1000!
antonfire confused me  I don't want to even know what he was doing with numerology.
Answer is 249 for reasons listed  that website is really good at explaining it.
LoopQuantumGravity  where were you able to multiply that out and get all the numbers. btw  my first guess before I looked into it was that it was well over 1000 zeros. I think I may have gotten excited and said 1,000,000 XD
Answer is 249 for reasons listed  that website is really good at explaining it.
LoopQuantumGravity  where were you able to multiply that out and get all the numbers. btw  my first guess before I looked into it was that it was well over 1000 zeros. I think I may have gotten excited and said 1,000,000 XD
 LoopQuantumGravity
 Posts: 416
 Joined: Mon Oct 01, 2007 4:19 am UTC
Re: 1000!
Gyvulys624 wrote:antonfire confused me  I don't want to even know what he was doing with numerology.
Answer is 249 for reasons listed  that website is really good at explaining it.
LoopQuantumGravity  where were you able to multiply that out and get all the numbers. btw  my first guess before I looked into it was that it was well over 1000 zeros. I think I may have gotten excited and said 1,000,000 XD
I used mathematica to do it for me. It's only a few thousand digits. 1000! ~ 4.02*10^2567.
I study theoretical physics & strings, and am a recipient of the prestigious Jayne Cobb Hero of Canton award.
And the science gets done and you make a neat gun
For the people who are still alive!
And the science gets done and you make a neat gun
For the people who are still alive!
Re: 1000!
LoopQuantumGravity wrote:Gyvulys624 wrote:antonfire confused me  I don't want to even know what he was doing with numerology.
Answer is 249 for reasons listed  that website is really good at explaining it.
LoopQuantumGravity  where were you able to multiply that out and get all the numbers. btw  my first guess before I looked into it was that it was well over 1000 zeros. I think I may have gotten excited and said 1,000,000 XD
I used mathematica to do it for me. It's only a few thousand digits. 1000! ~ 4.02*10^2567.
Yeah x! grows slower than most mathematicians intuitively feel it should, I've often found people trying to come up with crazy algorithms to try and reduce the size of the number they're dealing with when they have x!, this often just isn't needed.
I'm so geeky I got really excited about making this signature in SVG until it occurred to me HTML would obviously be turned off ¬_¬'
Re: 1000!
Dividing by four gives a good estimate since x/5+x/25+x/125...~=x/4
 Indigo is a lie.
Which idiot decided that websites can't go within 4cm of the edge of the screen?
There should be a null word, for the question "Is anybody there?" and to see if microphones are on.
Re: 1000!
Macbi wrote:Dividing by four gives a good estimate since x/5+x/25+x/125...~=x/4
Using the floor of each of those fractions is the method I was taught in number theory. 1037! factorial has 207 + 41 + 8 + 1 = 257 terminal zeroes.
Re: 1000!
Macbi wrote:Dividing by four gives a good estimate since x/5+x/25+x/125...~=x/4
Subtracting the sum of the digits in base 5 before dividing gives an even better estimate. In fact, it gives the correct answer. See if you can work out why this is true.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?
Re: 1000!
I can see why subtracting the base 5 digital sum first would give you the exact value, but I don't have anything written down. What is obvious is that you will get a multiple of 4 when you subtract the base 5 digital sum from the number, just like you get a multiple of 9 when you subtract the decimal digital sum from the number. So when you divide by 4 after doing that, you never get a fraction.
On a related note, if you count the number of zeros at the end of 36015!, IT'S OVER NINETHOUSAND!!1!one!!!!11! (it's 9001)
On a related note, if you count the number of zeros at the end of 36015!, IT'S OVER NINETHOUSAND!!1!one!!!!11! (it's 9001)
LOWA
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