Find the greatest number 23a68b, which is divisible by 3 but NOT divisible by 9.

The correct option is 239685.

We need to find the greatest number of the form 23a68b (where a and b are single-digit numbers) that is divisible by 3 but NOT divisible by 9, choosing from the given options.
Let us evaluate each of the given options to see if they match the form 23a68b and satisfy the divisibility criteria:

Option 1: 238689
This matches the form 23a68b with a = 8 and b = 9.
Let us calculate the sum of its digits to check for divisibility by 3 and 9:
Sum of digits = 2 + 3 + 8 + 6 + 8 + 9 = 36.
Since the sum of the digits (36) is divisible by 9 (and therefore also by 3), this number is divisible by 9. Thus, it does not satisfy the condition of NOT being divisible by 9.

Option 2: 239685
This matches the form 23a68b with a = 9 and b = 5.
Let us calculate the sum of its digits:
Sum of digits = 2 + 3 + 9 + 6 + 8 + 5 = 33.
Since the sum of the digits (33) is divisible by 3 but not divisible by 9, the number 239685 is divisible by 3 but NOT divisible by 9. This satisfies all requirements.

Option 3: 239688
This matches the form 23a68b with a = 9 and b = 8.
Let us calculate the sum of its digits:
Sum of digits = 2 + 3 + 9 + 6 + 8 + 8 = 36.
Since the sum of the digits (36) is divisible by 9, this number is divisible by 9, which violates the condition.

Option 4: 237687
This matches the form 23a68b with a = 7 and b = 7.
Let us calculate the sum of its digits:
Sum of digits = 2 + 3 + 7 + 6 + 8 + 7 = 33.
This number is divisible by 3 but not by 9. However, comparing it with 239685, we see that 239685 is larger (239685 > 237687).

Therefore, the greatest number matching the form that is divisible by 3 but not by 9 is 239685.