by 
ajinkya
ANSWERS: 3
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I won't answer your question, but I will tell you that 100! has 24 zeroes, and 150! has 37 zeroes. Good luck!
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You get a zero whenever a 2 and a 5 come together to make a factor of ten. Since factorials have loads of factors of 2 it depends on the factors of 5 which will come from 5, 30,55,80,105 (1 zero each) 10,35,60,85,110 (1 zero each) 15,40,65,90,115 (1 zero each) 20,45,70,95,120 (1 zero each) 25,50,75,100 (2 zeros each) That's 28 zeros. Next comes: 125 which has three zeros making 31 zeros. So NO factorial has exactly 30 zeros, but "125!" and greater have at least 31 I wish I could think of some way of testing if I'm right.
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You need a number with at least 30 factors of 5. let's count, 5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,80,85,90,95,100,105,110,115,120,125 If you actually counted you will say "Hey, there are only 25 numbers there, what's the deal?" Well some of the numbers there have more than one factor of 5 contained in them. For example, 25=5*5 and 125=5*5*5 25,50,75 and 100 all have two factors of 5, so that is four more 5's and 125 has three factors of 5, so that is two more. Now we can see that 125! has 31 factors of 5. "What good will that do us?" you may be asking. Well, 125! also happens to have way more factors of 2, and the prime factorization of 10 is 5 and 2: so each of those 5's from before can be multiplied with a factor of 2 to get 10. Thus 125! ends in 30 zeros (actually 31) The factorial of any number less than 125 won't end in 30 zeroes because the number 125 contains three factors of 5, dropping the zero count of 120! (or 121! 122! 123! 124!) down to only 28 zeroes. Any factorial greater than 125 factorial will also end in 30 or more zeroes.
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