COUNT THE NUMBER OF STUDENTS
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In a Olympiad 13 boys & d girls have participated.Each student has got same marks & their total number is $d^2+10d+17$. Find out the total number of students.
- Phlembac Adib Hasan
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Re: COUNT THE NUMBER OF STUDENTS
First thanks to Pritom Vaia for giving such a lovely problem.
Is it self-made?
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Re: COUNT THE NUMBER OF STUDENTS
In such problems try to use divisibility rather than congruence, though both are same. Because using the sign $|$ is more convenient and helpful, also makes a better sense I think. After all, the solution remains much clearer.Phlembac Adib Hasan wrote:First thanks to Pritom Vaia for giving such a lovely problem.Is it self-made?
Then you would have to write only: \[d+13|d^2+10d+17,d^2+13d\Rightarrow d+13|3d-17,3d+39\Rightarrow d+13|56\]
Doesn't it look better?
One one thing is neutral in the universe, that is $0$.
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Re: COUNT THE NUMBER OF STUDENTS
No....... it,s a problem form Kaikobad sir's book ....& I also solved the problem using function..... I think by this problem,one could be able to understand the real beauty of function ...
Re: COUNT THE NUMBER OF STUDENTS
$f(-13) \equiv 0$?MATHPRITOM wrote:No....... it,s a problem form Kaikobad sir's book ....& I also solved the problem using function..... I think by this problem,one could be able to understand the real beauty of function ...
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Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
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Re: COUNT THE NUMBER OF STUDENTS
No, since ,here d is the number of girls .. we can't think d= -13. here,each student got same point.so,the total score is a function of total students. It was my thought & then, I approached with my thought.*Mahi* wrote:$f(-13)=0$?MATHPRITOM wrote:No....... it,s a problem form Kaikobad sir's book ....& I also solved the problem using function..... I think by this problem,one could be able to understand the real beauty of function ...
Re: COUNT THE NUMBER OF STUDENTS
Residue of $f(x)$ while divided by $x-a$ is $f(a)$, again $P(x)=x^2+10x+17$ is divisible by $x+13$ so $x+13|f(x) \Rightarrow x+13|f(-13) =54$.MATHPRITOM wrote:No, since ,here d is the number of girls .. we can't think d= -13. here,each student got same point.so,the total score is a function of total students. It was my thought & then, I approached with my thought.*Mahi* wrote: $f(-13) \equiv 0$?
Please read Forum Guide and Rules before you post.
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi