1. **Problem Statement:** Find the LCM of two positive integers $p$ and $q$ where $p = ab^2$ and $q = a^3b$, with $a$ and $b$ being prime numbers. 2. **Formula and Rules:** The LCM of two numbers is found by taking the highest power of each prime factor appearing in the factorization of the numbers. 3. **Step-by-step Solution:** - Given $p = ab^2$ and $q = a^3b$. - Prime factorization of $p$ is $a^1 b^2$. - Prime factorization of $q$ is $a^3 b^1$. - For LCM, take the maximum power of each prime: - For $a$: max power is $3$ (from $q$). - For $b$: max power is $2$ (from $p$). - Therefore, $$\text{LCM}(p, q) = a^3 b^2$$ 4. **Final Answer:** The LCM of $p$ and $q$ is $a^3 b^2$. This corresponds to option (c).