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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