1. **State the problem:** We need to write a context-free grammar (CFG) for the language $L = \{a \mid a \text{ contains equal number of } x\text{'s and } y\text{'s and the number of } x\text{'s is a multiple of } 3\}$. 2. **Understand the requirements:** - The string must have equal numbers of $x$ and $y$. - The number of $x$ (and thus $y$) must be a multiple of 3. 3. **Key insight:** Since the number of $x$ and $y$ are equal and the number of $x$ is a multiple of 3, the total number of $x$ and $y$ pairs is a multiple of 3. 4. **Grammar construction:** - Let $S$ be the start symbol. - We generate strings with equal numbers of $x$ and $y$ in groups of 3 pairs. - Each group of 3 pairs can be generated by a nonterminal $G$ that produces strings with exactly 3 $x$'s and 3 $y$'s in any order but balanced. 5. **Define $G$:** $G$ generates strings with 3 $x$'s and 3 $y$'s and equal numbers of each. 6. **Define $S$:** $S \to GG | GGG | \epsilon$ to generate multiples of 3 pairs (0, 3, 6, ... pairs). 7. **Simplify:** We can define $S \to SS | G | \epsilon$ to generate any multiple of $G$. 8. **Define $G$ explicitly:** $G$ generates strings with 3 $x$'s and 3 $y$'s balanced. One way is to use a balanced grammar for equal numbers of $x$ and $y$ and restrict to length 6. 9. **Balanced pairs grammar:** $B \to xBy | yBx | BB | \epsilon$ generates equal numbers of $x$ and $y$. 10. **Restrict $G$ to length 6:** $G$ can be defined as all strings generated by $B$ of length 6. 11. **Final grammar:** - $S \to SS | G | \epsilon$ - $G \to x x x y y y$ in all balanced permutations (can be generated by $B$ with length 6) - $B \to xBy | yBx | BB | \epsilon$ This grammar generates strings with equal numbers of $x$ and $y$ where the number of $x$ is a multiple of 3. **Final answer:** $$ \begin{aligned} S &\to SS \mid G \mid \epsilon \\ G &\to \text{all balanced strings with 3 } x \text{'s and 3 } y \text{'s (generated by } B \text{ of length 6)} \\ B &\to xBy \mid yBx \mid BB \mid \epsilon \end{aligned} $$ This CFG satisfies the problem requirements.