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BdMO Online Forum • View topic - Find $x$

Find $x$

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Find $x$

Post Number:#1  Unread postby Absur Khan Siam » Sat Feb 04, 2017 10:39 pm

Find $x$ such that $x^2 - 3x + 7 \equiv 2 (mod 11)$
"(To Ptolemy I) There is no 'royal road' to geometry." - Euclid
Absur Khan Siam
 
Posts: 53
Joined: Tue Dec 08, 2015 4:25 pm
Location: Bashaboo , Dhaka

Re: Find $x$

Post Number:#2  Unread postby Absur Khan Siam » Sat Feb 04, 2017 10:47 pm

Check my solution:
This equation can be rewritten :
$x(x-3) \equiv -5 \equiv 6 (mod 11)$
But if we check the factors of $6$,we will find
that there is no solution of $x$. :(
Last edited by Absur Khan Siam on Sun Feb 05, 2017 10:09 pm, edited 1 time in total.
"(To Ptolemy I) There is no 'royal road' to geometry." - Euclid
Absur Khan Siam
 
Posts: 53
Joined: Tue Dec 08, 2015 4:25 pm
Location: Bashaboo , Dhaka

Re: Find $x$

Post Number:#3  Unread postby dshasan » Sun Feb 05, 2017 7:04 pm

$x = 11n + 7$ for any non-negative integer n.

This can be proved very easily. $x \equiv {0,1,2,3,4,5,6,7,8,9,10}$. Now checking the values for $x(x-3) mod 11$ gives the only solution when $x \equiv 7(mod 11)$.
The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.

- Charles Caleb Colton
dshasan
 
Posts: 66
Joined: Fri Aug 14, 2015 6:32 pm
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Re: Find $x$

Post Number:#4  Unread postby Absur Khan Siam » Sun Feb 05, 2017 10:12 pm

I missed the point.Thanks to dhasan.
Correct solution:
This equation can be rewritten :
$x(x-3) \equiv -5 \equiv 6 (mod 11)$
If we check the factors of $6$,we will find
that the solution is $x \equiv 7$. :D
"(To Ptolemy I) There is no 'royal road' to geometry." - Euclid
Absur Khan Siam
 
Posts: 53
Joined: Tue Dec 08, 2015 4:25 pm
Location: Bashaboo , Dhaka


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