BdMO National Higher Secondary 2007/6
Problem 6:
Writing down all the integers from $19$ to $92$ we make a large integer $N$.\[N=192021\cdots 909192\]If $N$ is divisible by $3^k$ then what is the maximum value of $k$?
Writing down all the integers from $19$ to $92$ we make a large integer $N$.\[N=192021\cdots 909192\]If $N$ is divisible by $3^k$ then what is the maximum value of $k$?
- bristy1588
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Re: BdMO National Higher Secondary 2007/6
Can anyone tell me how to proceed in these types of Problems?
Bristy Sikder
Re: BdMO National Higher Secondary 2007/6
Here it is to be used that if the sum of digits of $N$ is divisible by $3^k$, then so is $N$.BdMO wrote:Problem 6:
Writing down all the integers from $19$ to $92$ we make a large integer $N$.\[N=192021\cdots 909192\]If $N$ is divisible by $3^k$ then what is the maximum value of $k$?
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Re: BdMO National Higher Secondary 2007/6
Mahi, is it true for $k>2$ ?Of course, it can be used for checking that is $k<2$ or not. Because when the sum of all digit of "N" can divisible by 9 then $k$ can be greater or equal $2$.*Mahi* wrote: Here it is to be used that if the sum of digits of $N$ is divisible by $3^k$, then so is $N$.
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- nafistiham
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Re: BdMO National Higher Secondary 2007/6
$3^3=27$, but, $27$ does not divide $2+7=9$
\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]
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Re: BdMO National Higher Secondary 2007/6
This theorem can be improvised, like $1000 \equiv 1 mod 27$sourav das wrote:Mahi, is it true for $k>2$ ?Of course, it can be used for checking that is $k<2$ or not. Because when the sum of all digit of "N" can divisible by 9 then $k$ can be greater or equal $2$.*Mahi* wrote: Here it is to be used that if the sum of digits of $N$ is divisible by $3^k$, then so is $N$.
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Re: BdMO National Higher Secondary 2007/6
Actually, i just want to say we just can check whether N is divisible by 9 or not for interesting digit pattern and if satisfy, we can think other ways but not in digit way. It is easy to check that $N$ is not divisible by 9. As 19+20+....+91=37*111, which is only divisible by 3, not 9.
You spin my head right round right round,
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
When you go down, when you go down down......(-$from$ "$THE$ $UGLY$ $TRUTH$" )
Re: BdMO National Higher Secondary 2007/6
Thanks for opening your eyes.
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Nur Muhammad Shafiullah | Mahi
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