Problem 7:
How many positive prime numbers can be written as an alternating sequence of $1$'s and $0$'s where the first and last digit is $1$?
An alternating sequence of $1$'s and $0$'s is for example: $N = 1010101$ and has the property that $99N = 99999999$.
BdMO National Higher Secondary 2009/7
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
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Re: BdMO National Higher Secondary 2009/7
If the number has even number of 1's, it will be dividable by 101. But could be the next steps??
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Mehfuj Zahir
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Re: BdMO National Higher Secondary 2009/7
prove that 1010101.......is a geometric sequence.example10101=10000+100+1. now try to prove that this type of sequence has factor.so you will get one prime number is 101
Re: BdMO National Higher Secondary 2009/7
thank you for the suggestion.
Please Install $L^AT_EX$ fonts in your PC for better looking equations,
Learn how to write equations, and don't forget to read Forum Guide and Rules.
"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes
Learn how to write equations, and don't forget to read Forum Guide and Rules.
"When you have eliminated the impossible, whatever remains, however improbable, must be the truth." - Sherlock Holmes
Re: BdMO National Higher Secondary 2009/7
It is $10^{2k-2}+...+10^2+1=\frac {10^{2k}-1} {99}$
So if $k>1$ then let $k=2^rs,s$ odd and note that $x^{pq}-1$ is divisible by $(x^p-1)(x^q-1)$
So if $k>1$ then let $k=2^rs,s$ odd and note that $x^{pq}-1$ is divisible by $(x^p-1)(x^q-1)$
One one thing is neutral in the universe, that is $0$.