Step 1: Absence of absorbance at 450 nm rules out the yellow iso-alloxazine system of riboflavin-derived co-enzymes (FAD, FMN). The absorbance at 260 nm (A = 0.42, path-length 1 cm) gives

which matches the adenine chromophore present in nicotinamide nucleotides (NAD⁺/NADP⁺). Hence the co-enzyme is nicotinamide-based.
Step 2: Complete hydrolysis of 1.0 × 10⁻⁴ mol of C liberates 1.0 × 10⁻⁴ mol of a fragment that is stable to hot phosphoric acid, reduces Tollens’ reagent and Benedict’s solution → the fragment is a reducing sugar. In NADP⁺ the 2′-phosphate on the adenosine ribose is hydrolytically stable; after cleavage of the pyrophosphate and nicotinamide–ribose bonds this 2′-phospho-ribose (or ribose-5-phosphate) is released and behaves as a reducing sugar. Thus the extra phosphate of NADP⁺ is responsible for the observed reducing sugar.
Step 3: Exhaustive enzymatic de-phosphorylation (removes all phosphates) followed by acid hydrolysis of 1.0 × 10⁻⁴ mol C gives 2.0 × 10⁻⁴ mol NH₃ and 1.0 × 10⁻⁴ mol of a 6-carbon aldaric acid. Two mol NH₃ per mol C indicates two amino groups in the original molecule; these are the amide NH₂ of nicotinamide and the amino group of the adenine ring (both released on strong acid hydrolysis). The 6-carbon aldaric acid arises from the glucose/ribose skeleton. The acid is optically active and its meso-form is inactive → it must be an aldaric acid with a C2-axis, e.g. D-glucaric acid (obtained by nitric acid oxidation of D-glucose). L-idaric and L-gularic acids are not consistent with the symmetry argument.
Step 4: Combining the 260 nm absorbance, the extra phosphate (giving a reducing sugar on hydrolysis), and the formation of D-glucaric acid, the co-enzyme must be NADP⁺.
Conclusion: C is NADP⁺ and the 6-carbon aldaric acid is D-glucaric acid.