Thursday, 18 August 2016

Solving a basic linear congruence



I have been tasked with solving a linear congruence:



-12x\equiv-3\pmod{26}







How do I do this? I've never done linear congruences with minus signs so I'm quite confused.



Usually I would find the inverse of the LHS and multiply the RHS by the inverse however obviously since we have a negative number, the number isn't in \mathbb{Z}_{26} so we can't find an inverse.


Answer



For some integer k,
\begin{align*} -12x&\equiv -3 \pmod{26}\\ -12x &= -3 + 26k \end{align*}
The left hand side is divisible by 2, but the right hand side is not, so there is no solution for x.


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