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Math Help - equatle square root of huge number

  1. #1
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    equatle square root of huge number

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  2. #2
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    Quote Originally Posted by hotgal24 View Post
    *Ahem* 333...33 (1000 digits)
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  3. #3
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    Note that \underbrace{111\ldots111}_{\textrm{$n$ digits}}\ =\ \frac{10^n-1}{9}.

    Hence

    \underbrace{111\ldots111}_{\textrm{2$n$ digits}}-\underbrace{222\ldots222}_{\textrm{$n$ digits}}\ =\ \frac{10^{2n}-1}{9}\,-\,2\cdot\frac{10^n-1}{9}

    \color{white}.\hspace{42mm}. =\ \frac{\left(10^n+1\right)\left(10^n-1\right)}{9}-\frac{2\left(10^n-1\right)}{9}

    \color{white}.\hspace{42mm}. =\ \frac{\left(10^n-1\right)\left(10^n+1-2\right)}{9}

    \color{white}.\hspace{42mm}. =\ \left(\frac{10^n-1}{3}\right)^2

    Substitute n=1000 and you have your answer.
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  4. #4
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    Quote Originally Posted by mr fantastic View Post
    *Ahem* 333...33 (1000 digits)
    Some hints:

    222...22 = 2(111...11) (1000 digits)

    111...11 (2000 digits) divided by 111...11 (1000 digits) = 1000...001 (1001 digits)

    1000...001 - 2 = 999...99 = 9(111...11) (1000 digits)
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