Dhaka Higher Secondary 2010/6
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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.
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.
dhaka div 2010
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?
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?
 Tahmid Hasan
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Re: dhaka div 2010
Yeah! I had also got it $3$. I think it would be $mod 3$ instead of $mod 9$.
Btw who understood that $S_n(x)$
Btw who understood that $S_n(x)$
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Re: dhaka div 2010
i am not understanding .... . Going to PM moon bhaya
Re: dhaka div 2010
at first, $2^{6n+4}\equiv1 (mod 3)$
again, $5^{6n+4}\equiv1 (mod 3)$
so $2^{1024}+5^{1024}+1\equiv1+1+1=3 (mod 3)$
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
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: dhaka div 2010
The question in Dhaka was:
What is the remainder when $2^{1024} + 5^{1024}$ is divided by $3$?
The answer is $2$
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
Re: dhaka div 2010
that was for the H.secondary .Avik Roy wrote:The question in Dhaka was:
What is the remainder when $2^{1024} + 5^{1024}$ is divided by $3$?
The answer is $2$
the problem i gave was from the secondary group .
Re: dhaka div 2010
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.
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
Re: dhaka div 2010
oh thanks for the clarification . i was stumped to the answer .

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Re: dhaka div 2010
Ascha tushar tumi j rule a rule a math ta korle ata congruence na onno kishu .amake janao.pls