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Quant - Number of digits in a number

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Intern
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Joined: 28 Aug 2012
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Quant - Number of digits in a number [#permalink] New post 30 Sep 2012, 07:12
Hi,

Is there any shortcut to find out number of digits in a number say (4^34)(5^38) ?
Help much appreciated.

Thanks,
Weirdo
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Re: Quant - Number of digits in a number [#permalink] New post 30 Sep 2012, 13:01
Weirdo2989 wrote:
Hi,

Is there any shortcut to find out number of digits in a number say (4^34)(5^38) ?
Help much appreciated.

Thanks,
Weirdo


Not a shortcut, but this is how I would do it: I would find the positive integer \(n\) for which \(10^n<(4^{34})(5^{38})<10^{n+1}\).

\(4^{34}\cdot{5^{38}}=2^{68}\cdot{5^{38}}=10^{38}\cdot{2^{30}}\).
\(2^{30}=(2^{10})^3=1024^3>1000^3=10^9.\)
Also, \(2^{30}=2^{10}\cdot{2^{20}}=2^{10}\cdot4^{10}<2^{10}\cdot5^{10}=10^{10}\).

Our number is between \(10^{38+9}\) and \(10^{38+10}\), therefore we can conclude that it has \(48\) digits (\(10^{47}\) has \(48\) digits and \(10^{48}\) has \(49\) digits).
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Re: Quant - Number of digits in a number [#permalink] New post 30 Sep 2012, 23:16
Simply amazing.. Thanks a ton :)

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Re: Quant - Number of digits in a number   [#permalink] 30 Sep 2012, 23:16
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Quant - Number of digits in a number

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