USAJMO 2016

For discussing Olympiad Level Number Theory problems
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Kazi_Zareer
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USAJMO 2016

Unread post by Kazi_Zareer » Fri Aug 26, 2016 9:14 pm

Prove that there exists a positive integer $n < 10^6$ such that $5^n$ has six consecutive zeros in its decimal representation.

This problem was proposed by Evan Chen.
We cannot solve our problems with the same thinking we used when we create them.

joydip
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Joined: Tue May 17, 2016 11:52 am

Re: USAJMO 2016

Unread post by joydip » Sat Dec 10, 2016 11:49 am

$2^{20} \parallel 5^{2^{18}}-1$ .Then $5^{2^{18}+20} \equiv 5^{20} ( mod 10^{20})$. Now,$\lfloor {20\log 5} \rfloor+1=14$. So $ 5^{20}$ has 14 digits. So $5^{2^{18}+20}$ has exactly six consecutive zeros. Now ${2^{18}+20}< 10^6$ can be easily proved.
The first principle is that you must not fool yourself and you are the easiest person to fool.

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