Explanation:

- To find the greatest number that divides 150, 270, and 180 exactly, we need to calculate the greatest common divisor (GCD).

- The factors of each number are:

- 150: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150

- 270: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 54, 90, 135, 270

- 180: 1, 2, 3, 5, 6, 9, 10, 12, 15, 18, 30, 36, 45, 60, 90, 180

- The common divisors are: 1, 2, 3, 5, 6, 10, 15, 30

- Among these, the greatest common divisor is 30.

Therefore, the greatest number which divides 150, 270, and 180 exactly is Option 2: 30.