What is the probability that a randomly selected factor for a positive factors of 72 is less than 11?

Options

A. `(5)/(12)`
B. `(7)/(11)`
C. `(7)/(12)`
D. `(7)/(10)`

Answer

To find the probability that a randomly selected positive factor of 72 is less than 11, we will follow these steps:

### Step 1: Determine the positive factors of 72.
To find the factors of 72, we can start by finding its prime factorization:
- 72 can be expressed as \( 72 = 2^3 \times 3^2 \).

Using the prime factorization, we can find the total number of factors using the formula:
\[ \text{Total factors} = (e_1 + 1)(e_2 + 1) \ldots (e_n + 1) \]
where \( e_1, e_2, \ldots, e_n \) are the powers of the prime factors.

For 72:
- The powers are 3 (for 2) and 2 (for 3).
- Thus, the total number of factors is \( (3 + 1)(2 + 1) = 4 \times 3 = 12 \).

### Step 2: List all positive factors of 72.
The positive factors of 72 are:
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.

### Step 3: Identify the factors that are less than 11.
From the list of positive factors, we identify those that are less than 11:
1, 2, 3, 4, 6, 8, 9.

### Step 4: Count the number of factors less than 11.
The factors less than 11 are:
- 1, 2, 3, 4, 6, 8, 9.
- There are 7 factors in total.

### Step 5: Calculate the probability.
The probability \( P \) that a randomly selected factor of 72 is less than 11 is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\[ P = \frac{\text{Number of factors less than 11}}{\text{Total number of factors}} = \frac{7}{12}. \]

### Final Answer:
The probability that a randomly selected positive factor of 72 is less than 11 is \( \frac{7}{12} \).

---