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MATH- HELP

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

  • MATH- HELP -

    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.

  • MATH- HELP -

    Thank you!

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