Step 1: Instead of using the LCM directly, test each statement with concrete counter-example numbers to check whether the answer to "is $n$ divisible by 6?" stays consistent.
Step 2: Test Statement I alone: take $n=9$, which is divisible by 3. Is $9$ divisible by 6? No. Now take $n=6$, which is also divisible by 3. Is $6$ divisible by 6? Yes. Since Statement I allows both a "no" and a "yes" answer, Statement I alone is NOT sufficient.
Step 3: Test Statement II alone: take $n=4$, which is divisible by 4. Is $4$ divisible by 6? No. Now take $n=12$, which is also divisible by 4. Is $12$ divisible by 6? Yes. Since Statement II also allows both a "no" and a "yes" answer, Statement II alone is NOT sufficient.
Step 4: Test both statements together: $n$ must be divisible by both 3 and 4 simultaneously. Any such $n$ must be a multiple of $\text{lcm}(3,4)=12$, i.e. $n\in\{12,24,36,...\}$.
Step 5: Every multiple of 12 is automatically a multiple of 6 (since $12=2\times6$), so the answer to "is $n$ divisible by 6?" is consistently "yes" for every value satisfying both statements. Together, the statements ARE sufficient, while neither alone is.
\[\boxed{\text{Both statements together are sufficient, but neither statement alone is sufficient.}}\]
Correct option: (3) Both statements together are sufficient, but neither statement alone is sufficient.