how many zeros has the number 1000! at the end? and what is the last digit before these zeros?


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Question Number 134034 by mr W last updated on 27/Feb/21

$h o w m a n y z e r o s h a s t h e n u m b e r$ $1000! a t t h e e n d ? a n d w h a t i s t h e$ $l a s t d i g i t b e f o r e t h e s e z e r o s ?$
	Answered by floor(10²Eta[1]) last updated on 27/Feb/21

	$⌊ \frac{1000}{5} ⌋ + ⌊ \frac{1000}{5^{2}} ⌋ + ⌊ \frac{1000}{5^{3}} ⌋ + ⌊ \frac{1000}{5^{4}} ⌋$ $= 200 + 40 + 8 + 1$ $= 249 zeros$ $the last digit before the zeros is$ $\frac{1000!}{10^{249}} (\mod 10) = ?$
	Commented by mr W last updated on 27/Feb/21

	$100 h a s t w o t i m e s 10 . y o u c o u n t e d o n l y o n e t i m e .$ $1000 h a s t h r e e t i m e s 10 . y o u c o u n t e d o n l y o n e t i m e .$ $50, 150, 250 . . . e a c h f o r m s a 10 a n d a 5 . y o u c o u n t e d o n l y 10 .$
	Commented by mr W last updated on 27/Feb/21

	$249 z e r o s a r e c o r r e c t .$
	Commented by malwan last updated on 27/Feb/21

	$w h a t i s m i s s i n g i n m y m e t h o d s i r ?$
	Commented by malwan last updated on 28/Feb/21

	$y e s s i r$ $t h a n k y o u s o m u c h$
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