I'm trying to solve the following linear equation system using modular arithmetic with modulo 251. I know it can also be resolved using Gauss Jordan but I'm not sure how to do it applying modulo 251 in all operations. G1, G2, G3 and G4 are constants.
G1 = a + b + c +d
G2 = a + 2*b + 4*c + 8*d
G3 = a + 3*b + 9*c + 27*d
G4 = a + 4*b + 16*c + 64*d
I tried to solve the system but my solution seems to be wrong. Can anyone please give me the correct answer and way to solve this?
The system is given by $Ax=b$, and since $\det(A)=12$ is nonzero in $\Bbb F_{251}$, the solution is unique, namely $x=A^{-1}b$. Note that the determinant is of Vandermonde type.
Actually, in $\Bbb F_{251}$ we have $$ A^{-1}=\begin{pmatrix} 4 & 245 & 4 & 250\cr 163 & 135 & 244 & 211 \cr 127 & 247 & 129 & 250 \cr 209 & 126 & 125 & 42 \end{pmatrix} $$