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Without using a calculator, how can we solve the following?

  1. How do we find the number of zeros at the end of $600!$
  2. What are the last 3-digits of $171^{172}$?
  3. What is the sum of all positive numbers less than or equal to $61$, which are divisible by $3$ as well as by $5$?
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    The number of trailing zeros in a factorial has been discussed in http://math.stackexchange.com/questions/226868/number-of-trailing-zeros-in-a-factorial-in-base-b and http://math.stackexchange.com/questions/111385/derive-a-formula-to-find-the-number-of-trailing-zeroes-in-n and http://math.stackexchange.com/questions/17916/how-come-the-number-n-can-terminate-in-exactly-1-2-3-4-or-6-zeroes-but-n and http://math.stackexchange.com/questions/141196/highest-power-of-a-prime-p-dividing-n – 2012-11-12

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