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Math Help - Inclusion/exclusion principle for amount of numbers

  1. #1
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    Inclusion/exclusion principle for amount of numbers

    Hello, I came across this problem and I wasn't entirely sure on how to do it. Seeing if someone can give me a hand with it? Any help is greatly appreciated, thanks!
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  2. #2
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    Quote Originally Posted by Mahonroy View Post
    Hello, I came across this problem and I wasn't entirely sure on how to do it. Seeing if someone can give me a hand with it? Any help is greatly appreciated, thanks!

    Let's call n_2\,,\,\,n_3 to the number of squares (cubes) less than 1,000:

    10^3=1,000\Longrightarrow n_3=10

    31^2=961\,,\,32^2=1,024>1,000\Longrightarrow n_3=31.

    Now, just take out ONCE the repeated ones: for example, 64=4^3=8^2, so we must erase 64 since it appears twice...

    Tonio
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    Thanks for the reply! I found the answer to be 38 is this what you got?
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  4. #4
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    Yes thats right.

    An easy way to see this is all perfect squares are of the form k^2, all perfect cubes are of the form k^3 and all numbers that are both a perfect square and a perfect cube are k^6. Using inclusion exculsion, that means for any  N the number of numbers that are a perfect square or a perfect cube less than N are  \lfloor N^{\frac{1}{2}} \rfloor + \lfloor N^{\frac{1}{3}} \rfloor - \lfloor N^{\frac{1}{6}} \rfloor
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