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Let, $\ n$ be the smallest positive number, such that:

the number, $\ S=8^n5^{600}$ has 604 digits

What is the sum of the digits?

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closed as off-topic by Trevor Wilson, Daniel Robert-Nicoud, Old John, user61527, Thomas Andrews Nov 29 '13 at 18:41

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  • 2
    $\begingroup$ what have you tried so far? what made you to think that this is important for you? $\endgroup$ – user87543 Nov 29 '13 at 15:42
  • $\begingroup$ Actually I was having problem with 5^600, I was wondering the length of that number....I should have thought in 10 ^600 terms. Rest of the portion will become easy to think then. Thanks again to all of you & stack exchange for your support. $\endgroup$ – Sourav Nov 30 '13 at 3:31
  • $\begingroup$ you have not responded to any of the answers... what would then be the use of thinking of helping you (from the context of users who answered this)... you should have been more responsible.... $\endgroup$ – user87543 Nov 30 '13 at 7:32
  • $\begingroup$ @PraphullaKoushik: It looked to me like you were OP. Sorry. $\endgroup$ – Ross Millikan Nov 30 '13 at 15:45
  • $\begingroup$ @RossMillikan : It is alright sir,there is nothing to say sorry ... you have helped me a lot.. I like this question.... $\endgroup$ – user87543 Nov 30 '13 at 16:20
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If $n=200$ then $S=8^{200}\cdot 5^{600} = 2^{600}\cdot 5^{600} = 10^{600}$ that has $601$ digits.

so if $n=204$ you have $S=8^4\cdot 10^{600} =4096\cdot 10^{600}$. It has $604$ digits and the sum is $4+9+6=19$.

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Hint: how many digits does $10^{600}$ have? How does this help?

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  • $\begingroup$ I am not at all familiar with this kind of questions but your hint made me to try something.... $10^{600}$ has $601$ digits... but how do i go further. please suggest something.. $\endgroup$ – user87543 Nov 29 '13 at 15:57
  • $\begingroup$ Do you see how to get $10^{600}$? Now you have to multiply it by something to get three more. All you have to play with is $8^n$ $\endgroup$ – Ross Millikan Nov 29 '13 at 16:38
  • $\begingroup$ yes... I could see that $10^=100$ has $3$ digits, $10^3=1000$ has $4$ digits in similar manner $10^{600} $ would have $601$ digits... $\endgroup$ – user87543 Nov 29 '13 at 16:51
  • $\begingroup$ That is correct. A particular $n$ will give you $8^n5^{600}=10^{600}$ Now what happens if you increase $n$ by $1$? By $2$? What are the new products? $\endgroup$ – Ross Millikan Nov 29 '13 at 16:53
  • $\begingroup$ I need $8^n=2^{600}$ then i would have $8^n 5^{600}=2^{600}5^{600}=10^{600}$... So i need $8^n=2^{600}$ i.e., $2^{3n}=2^{600}$ this implies $n=200$... $\endgroup$ – user87543 Nov 30 '13 at 7:42
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The condition that $8^n\cdot 5^{600}$ has $604$ digits can be written $$ \log_{10}(8^n\cdot 5^{600})\ge 603 $$ or $$ 3n\log_{10}2+600(1-\log_{10}2)\ge 603, $$ that is, $$ 3n\log_{10}2\ge3+600\log_{10}2 $$ or $$ n\ge\frac{1}{\log_{10}2}+200=\log_{2}10+200, $$ so the minimum $n$ is $204$, because $3<\log_{2}10<4$.

Thus $S=8^{4}\cdot10^{200}$.

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