Correct Answer: (C)

1. Analysis of Statement (1): The LCM of $n$ and $12$ is $60$. Prime factorization of $12 = 2^2 \times 3^1$ and $60 = 2^2 \times 3^1 \times 5^1$. For the LCM to be $60$, $n$ must be a factor of $60$ and must contribute the factor $5^1$ which is missing in $12$. Possible values for $n$ include $5, 10, 15, 20, 30, 60$. Since there are multiple values, (1) alone is not sufficient.

2. Analysis of Statement (2): The GCD of $n$ and $12$ is $1$. This means $n$ is coprime to $12$, implying $n$ is not divisible by $2$ or $3$. There are infinitely many positive integers that satisfy this (e.g., $1, 5, 7, 11, \dots$). Thus, (2) alone is not sufficient.

3. Combining Both Statements: From (1), $n \in \{5, 10, 15, 20, 30, 60\}$. From (2), $n$ cannot be divisible by $2$ or $3$. Let us check the candidates: $5$ (not divisible by 2 or 3 - OK), $10$ (divisible by 2 - NO), $15$ (divisible by 3 - NO), $20$ (divisible by 2 - NO), $30$ (divisible by 2 and 3 - NO), $60$ (divisible by 2 and 3 - NO).

4. Conclusion: Only $n = 5$ satisfies both conditions. Therefore, the statements together provide a unique value.

Test Prep Tip: When dealing with LCM and GCD, always use prime factorization. Remember that $LCM(a, b) \times GCD(a, b) = a \times b$. In Data Sufficiency, once you narrow down to a single possible value using both statements, you have achieved sufficiency.