# Linear Congruence

• Sep 27th 2007, 04:03 PM
clockingly
Linear Congruence
The following was confusing me, so any help/explanations would be greatly appreciated!

Find all incongruent solutions to the following linear congruence:

21x ≡ 14 (mod 91)
• Sep 29th 2007, 04:12 PM
ThePerfectHacker
Quote:

Originally Posted by clockingly
The following was confusing me, so any help/explanations would be greatly appreciated!

Find all incongruent solutions to the following linear congruence:

21x ≡ 14 (mod 91)

$\displaystyle \gcd(21,91)=7$ and $\displaystyle 7|14$ so there shall be $\displaystyle 7$ solutions in each congruence class. By inspection we see that $\displaystyle x\equiv 5$ is one equivalence class which solves this. By theorem it means $\displaystyle 5 + \frac{91}{7}t$ for $\displaystyle t=0,1,...,6$ shall all be solutions.
• Sep 30th 2007, 07:07 AM
topsquark
Quote:

Originally Posted by ThePerfectHacker
$\displaystyle \gcd(21,91)=7$ and $\displaystyle 7|14$ so there shall be $\displaystyle 7$ solutions in each congruence class. By inspection we see that $\displaystyle x\equiv 5$ is one equivalence class which solves this. By theorem it means $\displaystyle 5 + \frac{91}{7}t$ for $\displaystyle t=0,1,...,6$ shall all be solutions.

By "incongruent" solutions the question is asking for all distinct residue classes that solve the equation?

-Dan
• Sep 30th 2007, 08:05 AM
ThePerfectHacker
Quote:

Originally Posted by topsquark
By "incongruent" solutions the question is asking for all distinct residue classes that solve the equation?

-Dan

Yes.

For example $\displaystyle x\equiv 2 (\bmod 4)$ the solutions $\displaystyle x=2,6$ are really the same ("congruent") solutions because they are contained in the same equivalence class (equivalence class mod $\displaystyle 4$*)

*)The congruence classes are $\displaystyle \{...,-4,0,4,...\} , \{...,-3,1,5,...\}, \{...,-2,2,6,...\}, \{...,-1,3,7,...\}$. So $\displaystyle x=2,6$ are in the same equivalence class.
• Jun 3rd 2008, 10:04 AM
duggaboy
hmm
so if showing the incongruences would it be x=-12=14 (mod13)??