Tuesday 29 May 2012

Without using a calculator, how many zeros are at the end of "100!"? (that's 100*99*98*...*3*2*1)

Answer: What you don't want to do is start multiplying it all out! The trick is
remembering that the number of zeros at the end of a number is equal to the
number of times "10" (or "2*5") appears when you factor the number. Therefore
think about the prime factorization of 100! and how many 2s and 5s there are.
There are a bunch more 2s than 5s, so the number of 5s is also the number of 10s in
the factorization. There is one 5 for every factor of 5 in our factorial multiplication
(1*2*...*5*...*10*...*15*...) and an extra 5 for 25, 50, 75, and 100. Therefore we have
20+4 = 24 zeros at the end of 100!.

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