Find the probability that a randomly chosen three-digit number has exactly three factors.

Options

A. `2//225`
B. `7//900`
C. `1//800`
D. none of these

Explanation

To find the probability that a randomly chosen three-digit number has exactly three factors, we can follow these steps:

### Step 1: Understanding the Condition for Exactly Three Factors
A number has exactly three factors if and only if it is the square of a prime number. This is because if \( p \) is a prime number, then \( p^2 \) has the factors \( 1, p, p^2 \), totaling three factors.

### Step 2: Identify the Range of Three-Digit Numbers
Three-digit numbers range from 100 to 999.

### Step 3: Find the Prime Numbers Whose Squares are Three-Digit Numbers
To find the three-digit numbers that are squares of prime numbers, we need to determine the prime numbers \( p \) such that \( p^2 \) falls within the range of 100 to 999.

- The smallest integer \( p \) such that \( p^2 \geq 100 \) is \( p = 10 \) (since \( 10^2 = 100 \)).
- The largest integer \( p \) such that \( p^2 \leq 999 \) is \( p = 31 \) (since \( 31^2 = 961 \) and \( 32^2 = 1024 \) which is a four-digit number).

### Step 4: List the Prime Numbers Between 10 and 31
The prime numbers between 10 and 31 are:
- 11
- 13
- 17
- 19
- 23
- 29
- 31

### Step 5: Calculate the Squares of These Prime Numbers
Now we calculate the squares of these prime numbers:
- \( 11^2 = 121 \)
- \( 13^2 = 169 \)
- \( 17^2 = 289 \)
- \( 19^2 = 361 \)
- \( 23^2 = 529 \)
- \( 29^2 = 841 \)
- \( 31^2 = 961 \)

This gives us a total of 7 three-digit numbers that have exactly three factors.

### Step 6: Determine the Total Number of Three-Digit Numbers
The total number of three-digit numbers is:
- From 100 to 999, there are \( 999 - 100 + 1 = 900 \) three-digit numbers.

### Step 7: Calculate the Probability
The probability \( P \) that a randomly chosen three-digit number has exactly three factors is given by the ratio of the favorable outcomes to the total outcomes:
\[ P = \frac{\text{Number of three-digit numbers with exactly three factors}}{\text{Total number of three-digit numbers}} = \frac{7}{900} \]

### Final Answer
Thus, the probability that a randomly chosen three-digit number has exactly three factors is \( \frac{7}{900} \).

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