Dhaka Higher Secondary 2010/6

Problem for Higher Secondary Group from Divisional Mathematical Olympiad will be solved here.
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Please don't post problems (by starting a topic) in the "Higher Secondary: Solved" forum. This forum is only for showcasing the problems for the convenience of the users. You can post the problems in the main Divisional Math Olympiad forum. Later we shall move that topic with proper formatting, and post in the resource section.
tushar7
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dhaka div 2010

Unread post by tushar7 » Tue Dec 28, 2010 2:43 pm

what is the remainder when $2^{1024} +5^{1024} +1$ is divided by $9$?
the answer given in KMC website is 0 . but i am having the same answer $3$
$2^3\equiv -1\bmod{9}$ $\Rightarrow 2^{1024}\equiv -2\bmod{9}$
$5^6\equiv 1\bmod{9}$ $\Rightarrow 5^{1024}\equiv 4\bmod{9}$
so $-2+4+1=3$ , where is the faulty?

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Tahmid Hasan
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Re: dhaka div 2010

Unread post by Tahmid Hasan » Tue Dec 28, 2010 6:37 pm

my ans is also 3!!!!!!!!! :shock:
বড় ভালবাসি তোমায়,মা

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Labib
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Re: dhaka div 2010

Unread post by Labib » Tue Dec 28, 2010 11:20 pm

Yeah! I had also got it $3$. I think it would be $mod 3$ instead of $mod 9$.
Btw who understood that $S_n(x)$
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tushar7
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Re: dhaka div 2010

Unread post by tushar7 » Tue Dec 28, 2010 11:28 pm

i am not understanding .... . Going to PM moon bhaya

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Labib
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Re: dhaka div 2010

Unread post by Labib » Tue Dec 28, 2010 11:37 pm

at first, $2^{6n+4}\equiv1 (mod 3)$
again, $5^{6n+4}\equiv1 (mod 3)$
so $2^{1024}+5^{1024}+1\equiv1+1+1=3 (mod 3)$
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

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Avik Roy
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Re: dhaka div 2010

Unread post by Avik Roy » Wed Dec 29, 2010 1:08 am

The question in Dhaka was:
What is the remainder when $2^{1024} + 5^{1024}$ is divided by $3$?
The answer is $2$
"Je le vois, mais je ne le crois pas!" - Georg Ferdinand Ludwig Philipp Cantor

tushar7
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Re: dhaka div 2010

Unread post by tushar7 » Wed Dec 29, 2010 1:15 am

Avik Roy wrote:The question in Dhaka was:
What is the remainder when $2^{1024} + 5^{1024}$ is divided by $3$?
The answer is $2$
that was for the H.secondary .
the problem i gave was from the secondary group .

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Avik Roy
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Re: dhaka div 2010

Unread post by Avik Roy » Wed Dec 29, 2010 1:18 am

sorry, I just noticed that this was a question in the secondary category....
The answer is $3$, if all of us are not mistaken.
However, there were a few corrections when the scripts were checked. So if the official solution was initially misprinted, it was corrected later.
"Je le vois, mais je ne le crois pas!" - Georg Ferdinand Ludwig Philipp Cantor

tushar7
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Re: dhaka div 2010

Unread post by tushar7 » Wed Dec 29, 2010 1:44 am

oh thanks for the clarification . i was stumped to the answer .

atiqur_jhe
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Re: dhaka div 2010

Unread post by atiqur_jhe » Fri Dec 31, 2010 9:14 pm

Ascha tushar tumi j rule a rule a math ta korle ata congruence na onno kishu .amake janao.pls

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