1,000,000,000!
How many 0s (zeroes) are there at the right hand end of 1,000,000,000! ?
That's one billion factorial ≈ 1.7 x 10^(20,723,265,847).
That's one billion factorial ≈ 1.7 x 10^(20,723,265,847).
Labels: mathschallenge
posted by Chris at
7:52 PM
15 Comments:
infinetally many
It's infinitely less than that many.
couldn't be as silly as "one" ... could it?
No, but I admire your courage suggesting it, LOL. It's not that sort of a trick question.
9
0
This post has been removed by the author.
sry that would be 9 + 8 + 7 + 6 + 5 + 4 + 3 +2 + 1 or 45
Think before you post I'm wrong
Sorry, but that's much too low. I've amended the problem to make it clear that it's 1 billion factorial we're dealing with.
But I am intrigued by your number.
ok so a 0 is added to the end of a factorial any time the factorization of the numbers includes a 2 and a 5 (the prime factorization of 10) There are considerably more numbers with 2 as a factor then with 5 as a factor so all w need to consider is the number of 5's that are in the prime factorization of 1000000000!. This can be given as 10^9/5 + 10^9/5^2 + 10^9/5^3 .... 10^9/5^12. At this point we can stop because 5^13 is larger then 10^9. With the division we only count the whole part because we are not interested in parts of 5's. This gives 200000000+40000000+8000000+1600000+320000+64000+12800+2560+512+102+20+4. So there are 249999998 0's at the end of 1000000000!
Bingo. David and Kimmy have got it. Well done, and thank you for your nice explanation. As you've probably noticed, your calculation is too wide for the page. Here are the numbers:
200,000,000 + 40,000,000 + 8,000,000 + 1,600,000 + 320,000
+ 64,000 + 12,800 + 2,560 + 512 + 102.x + 20.x + 4.x
The x's are unimportant as they correspond to numbers greater than 1,000,000,000.
249999998 ... lol .. factorials are good sleep medicine ... lol .. oh hell.. nm someone already answered... doh.. well. CHEERS!
I got a different answer. If it's wrong can someone point out why please.
Only factors of ten add zeroes.
If you multiply by ten you add one zero, if by one hundred you add 2 zeroes, if by a thousands 3 zeroes and so on.
Below is each different order of magnitude, the number of times it appears and the number of zeroes it adds(the pattern soon becomes apparent). the answer is quite nice too.
1,000,000,000 appears 1 time; its occurrence adds 9 zeroes
100,000,000 (or multiples thereof) appear 9 times; each occurrences adds 8 zeroes; total zeroes added is 9x8 = 72
Carrying on; the first number is order of magnitude; seconde is number of times a number of that order of magntitude occers; third is number of zeroes each adds and the last number is the total number of zeroes;
1,000,000,000; 1x9 =9.
100,000,000;9x8 = 72.
10,000,000; 90x7 = 630.
1,000,000; 900x6 = 5400.
100,000; 9000x5 = 45,000.
10,000; 90,000x4= 360,000.
1,000; 900,000x3= 2,700,000.
100; 9,000,000x2 = 18,000,000.
10; 90,000,000x1 = 90,000,000.
To get the total number of zeroes sum the right hand column which gives: 111,111,111. Pretty neat.
Hi Mister Fahrenheit. Try 2*5. No obvious 10 in it, so you would have missed it. Don't feel bad, it hurt my brain doing it.
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