Let's solve the problem step by step to determine which expression among the options provided is divisible by 6.

Step 1: Understanding the conditions for divisibility by 6

For a number to be divisible by 6, it must be divisible by both 2 and 3. Let's analyze each case with the given criteria.

Step 2: Analyze the given condition

We are told that \(n\) is a single-digit number greater than 2, divisible by 2, but not divisible by 4. Therefore, possible values for \(n\) are 2, 6, 10, ... However, since 6 is a multiple of 4, the only suitable candidates are 2 and 6.

Upon eliminating non-single digit numbers and ensuring divisibility by 2 and not 4, the possible values of \(n\) are 2 and 6, but since it cannot be divisible by 4, the valid number \(n\) is 6.

Step 3: Evaluating options

Using \(n = 6\), examine each expression:

• Option 1: 9n + 4
  Substituting \(n = 6\) gives: \(9 \times 6 + 4 = 54 + 4 = 58\). This is not divisible by 6.
• Option 2: 6n + 7
  Substituting \(n = 6\) gives: \(6 \times 6 + 7 = 36 + 7 = 43\). This is not divisible by 6.
• Option 3: 9n
  Substituting \(n = 6\) gives: \(9 \times 6 = 54\). This is divisible by 6.
• Option 4: 5n + 3
  Substituting \(n = 6\) gives: \(5 \times 6 + 3 = 30 + 3 = 33\). This is not divisible by 6.

Conclusion:

Upon checking each option with \(n = 6\), the expression \(9n\) is the only one divisible by 6. Hence, the correct answer is 9n.