 Posers and Puzzles

1. 24 Apr '14 07:43
How many zero's are at the end of 1 x 2 x 3 x .... x 1,000 ?

What's the non-zero number right before them?
2. 24 Apr '14 11:061 edit
Originally posted by talzamir
How many zero's are at the end of 1 x 2 x 3 x .... x 1,000 ?

What's the non-zero number right before them?
1000 factorial. I can't even get an answer in my HP calculator, it only goes up to 500!.

I assume therefore there is a shortcut, but no idea what that would be.

My casio poops out at 69!, which is 1.7 E98, so only up to 2 digit exponent and my HP I think only has 3 digit exponents.

So the answer must be 248 zero's.

I cheated🙂

http://voices.yahoo.com/the-number-zeros-end-1000-1000-factorial-888411.html
3. 24 Apr '14 12:111 edit
Kinda funny that the answer you cheated from is wrong. ^_^

The principle is right, of course. The zeros at the end come from the 5's and the 2's when 1000 ! is factored, and since there are more 2's than 5's, it's enough to count the fives.

Yahoo noted correctly that every fifth number is divisible by five, so there are 200 fives. Every fifth of those is a multiple of 5^2 = 25 so those count twice; add 40 fives. Every fifth of those is a multiple of 5^3 = 125 so those count three times, add 8 more fives, for a total of 200 + 40 + 8 = 248, as yahoo claimed.

But they missed that 5^4 = 625 which is less than 1,000, which gives one more, for a total of 249, not 248. :-P

So.. what's the last non-zero number before the zeros start?
4. 24 Apr '14 16:28
Originally posted by talzamir
Kinda funny that the answer you cheated from is wrong. ^_^

The principle is right, of course. The zeros at the end come from the 5's and the 2's when 1000 ! is factored, and since there are more 2's than 5's, it's enough to count the fives.

Yahoo noted correctly that every fifth number is divisible by five, so there are 200 fives. Every fifth of those ...[text shortened]... for a total of 249, not 248. :-P

So.. what's the last non-zero number before the zeros start?
Can't trust anyone anymore🙂
How many digits total are there in 1000!?
5. 24 Apr '14 21:222 edits
There are 2,568 of them as 1,000! = 10^2,567.605 = 4.02387 x 10^2,567.

The number exists online, but was simply enough to check with excel by taking a 10-base log of 1..1000 and adding them together.
6. 29 May '14 17:27
Originally posted by talzamir
There are 2,568 of them as 1,000! = 10^2,567.605 = 4.02387 x 10^2,567.

The number exists online, but was simply enough to check with excel by taking a 10-base log of 1..1000 and adding them together.