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Math Help - shortcut???????

  1. #1
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    shortcut???????

    what would be the no of digits in 461+462+463+464+465
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  2. #2
    MHF Contributor

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    Re: shortcut???????

    All of those numbers is less than or equal to 4^{65} so their sum is less than 5(4^{65}). 65 log(4)= 39.1339 so 4^{65} has 39 digits and 5 times that has 40 digits.
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  3. #3
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    Re: shortcut???????

    Hello, vjdm!

    \text{What is the number of digits in: }\:N \;=\;4^{61}+4^{62}+4^{63}+4^{64}+4^{65}

    We have: . N \;=\;4^{61}(1 + 4 + 4^2+4^3+4^4) \;=\;4^{61}\cdot341

    Take logs: . \log N \;=\;\log\left(4^{61}\cdot341\right) \;=\;\log\left(4^{61}\right) + \log(341)

    m . . . . . . . \log N \;=\;61\log(4) + \log(341) \;=\;39.25841385

    . . . . . . . . . . . N \;=\;1.813066986 \times 10^{39}


    Therefore, N has 40 digits.
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