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Thread: 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 $\displaystyle n_2\,,\,\,n_3$ to the number of squares (cubes) less than 1,000:

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

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

    Now, just take out ONCE the repeated ones: for example, $\displaystyle 64=4^3=8^2$, 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 $\displaystyle k^2$, all perfect cubes are of the form $\displaystyle k^3$ and all numbers that are both a perfect square and a perfect cube are $\displaystyle k^6$. Using inclusion exculsion, that means for any $\displaystyle N $ the number of numbers that are a perfect square or a perfect cube less than N are $\displaystyle \lfloor N^{\frac{1}{2}} \rfloor + \lfloor N^{\frac{1}{3}} \rfloor - \lfloor N^{\frac{1}{6}} \rfloor$
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