Venkaiah in the area of Error-free Computation Using Multiple-modulus Residue Arithmetic for Equivalent Symmetric Matrices and Linear Optimization around 1990.
2.
The whole point of the residue arithmetic is that the residues completely fail to imply which of the equivalent numbers is under discussion, so I don't think they'd yield anything about its evenness either.
3.
Venkaiah, Computing a matrix symmetrizer exactly using modified multiple modulus residue arithmetic, J . Computational and Applied Mathematics ( Elsevier Science Publishers B . V ., North-Holland ), 21, 1988, 27 40.
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