TITLE: How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?
QUESTION [44 upvotes]: Possible Duplicate:
Highest power of a prime $p$ dividing $N!$ 

How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?

REPLY [7 votes]: HINT $\: $ The power of a prime $\rm\:p\:$ dividing $\rm\ n!\:$ jumps from $\rm\: p-1\:$ for $\rm\: n = p^2-1\:$ to $\rm\: p+1\: $ for $\rm\: n = p^2\:$ since their are $\rm\:p-1\:$ naturals $\rm < p^2\ $ divisible by $\rm\:p\:,\:$ viz. $\rm\ p,\ 2\:p,\:\cdots\:,\: (p-1)\:p\:.\ $ Now put $\rm\ p = 5\:.$