If you multiply 100!*200!*300!, how many 0s are there?

Answer 1

Think of all the factors that will produce a zero

1. clearly all multiples of 10 will add a zero
that is, numbers like 30, 60, 210, etc

2. also every factor ending in 2 multiplied by a factor ending in 5 will produce a zero

so zeros are produced by the following
* 2 * * 5 * * * * 10 * 12 * * 15 * * * * 20 * ....

so for the first 10! we should have 2 zeros, test with your calculator to get 3628800
by the time we get to 20! we should have 4 zeros
at 30! we should have 6 zeros
etc
so after 100! we should have 20 zeros

by the same reasoning:
200! should have 40 zeros
300! should have 60 zeros

so we have (a x 10^20)(b x 10^40)(c x 10^60)
= abc x 10^120

check my thinking.

Answer 2

Thank you!