Step 1: Understanding the Concept
Williamson synthesis prepares ethers by reacting a sodium alkoxide with an alkyl halide: \(\text{R-ONa}+\text{R'-X}\rightarrow \text{R-O-R'}+\text{NaX}\).
Step 2: Apply to the reaction
The reagents are sodium 3-ethylpentan-3-oxide and \(\text{CH}_{3}\text{Br}\).
The alkoxide displaces bromide from the methyl carbon and gives 3-ethyl-3-methoxypentane and NaBr.
Step 3: Why the other options fail
Gattermann-Koch and Etard reactions prepare aldehydes.
The continuous etherification process uses an alcohol and conc. \(\text{H}_{2}\text{SO}_{4}\).
Final Answer:
The reaction is the Williamson synthesis, option (A).
\[ \boxed{\text{(A) }\text{Williamson synthesis}} \]