## Dhaka Higher Secondary 2010/6

Problem for Higher Secondary Group from Divisional Mathematical Olympiad will be solved here.
Forum rules
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
Posts: 101
Joined: Tue Dec 07, 2010 3:23 pm

### 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?

Tahmid Hasan
Posts: 665
Joined: Thu Dec 09, 2010 5:34 pm

### Re: dhaka div 2010

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

Labib
Posts: 411
Joined: Thu Dec 09, 2010 10:58 pm

### 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 $S_n(x)$
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tushar7
Posts: 101
Joined: Tue Dec 07, 2010 3:23 pm

### Re: dhaka div 2010

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

Labib
Posts: 411
Joined: Thu Dec 09, 2010 10:58 pm

### Re: dhaka div 2010

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

Avik Roy
Posts: 156
Joined: Tue Dec 07, 2010 2:07 am

### 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 $2$
"Je le vois, mais je ne le crois pas!" - Georg Ferdinand Ludwig Philipp Cantor

tushar7
Posts: 101
Joined: Tue Dec 07, 2010 3:23 pm

### Re: dhaka div 2010

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 .

Avik Roy
Posts: 156
Joined: Tue Dec 07, 2010 2:07 am

### 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.
"Je le vois, mais je ne le crois pas!" - Georg Ferdinand Ludwig Philipp Cantor

tushar7
Posts: 101
Joined: Tue Dec 07, 2010 3:23 pm

### Re: dhaka div 2010

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

atiqur_jhe
Posts: 13
Joined: Tue Dec 14, 2010 11:49 am
Location: Jhenidah

### Re: dhaka div 2010

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