To understand why each prime factor appears three times in its cube, let's consider a prime factor p.

**Prime Factorization**
The prime factorization of a number is the representation of that number as a product of its prime factors. For example, the prime factorization of 24 is 2 * 2 * 2 * 3, where 2 and 3 are prime factors.

**Cubing a Number**
When we cube a number, we multiply it by itself three times. For example, if we cube 2, we get 2 * 2 * 2 = 8.

**Prime Factorization of a Cubed Number**
Let's consider a prime factor p and its cube p^3. We can write p^3 as p * p * p.

Now, let's consider the prime factorization of p^3. Since p^3 is p * p * p, we can say that p^3 has three instances of the prime factor p.

Since p can be any prime number, this means that each prime factor appears three times in its cube.

**Example**
Let's take the prime factor 2 as an example.

The prime factorization of 2^3 (2 cubed) is 2 * 2 * 2, which has three instances of the prime factor 2.

Similarly, for any other prime factor p, p^3 will have three instances of the prime factor p.

Therefore, the correct answer is option B) 3, as each prime factor appears three times in its cube.