We consider two methods of Euclid's algorithm to solve the linear feedback shift register (LFSR) synthesis problem. One of the methods is identically equivalent to the celebrated Berlekamp-Massey (B-M) algorithm. The other method is distinctly Euclidean. The formulation of the problem from Euclid's algorithm leads to the characterization of the LFSR synthesis for the reverse sequence given the LFSR synthesis for the forward sequence.