Solution:

Step 1: Analyse Statement 1:The LCM of\(a\) and \(b\) is \(16\).

- Both a and b can have more than 1 combination for their LCM to be \(16\).
- For example, all the following combinations will have \(LCM (a, b) = 16\).

From the table, it is evident that there are more than 1 possible combinations of \(a\) and \(b\) where their LCM is \(16\). Hence, their product is also not unique.

Since this statement doesn’t give a unique answer, it is alone not sufficient to answer our question.

Statement 1 alone is NOT sufficient to answer the question.

Hence, we can eliminate answer choices A and D.

Step 2: Analyse Statement 2:The HCF of \(a\) and \(b\) is \(4\).

- Both a and b can have more than 1 combination for their GCD to be \(4\).
- For example, all the following combinations will have \(GCD (a, b) = 4\).

From the table, it is evident that there are more than 1 possible combinations of a and b where their GCD is 4, and hence, their product is also not unique.

Since this statement doesn’t give a unique answer, it is alone not sufficient to answer our question.

Step 3: Combine both Statements:- From the first Statement we got: \(LCM (a, b)\) = \(16\)
- From the second Statement we got:\(GCD (a, b)\) = \(4\)

Per our conceptual knowledge, we can say that the product of two numbers is the product of their LCM and GCD.

- Hence, \(a * b\)= \(LCM (a, b) * GCD (a, b)\)
- Or, \(a*b\) = \(16*4\) = \(64\)

By combining both statements we got a unique answer.

Correct Answer: Option C

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