The probability that a randomly chosen 3-digit number has exaclty 3 factors

Options

A. `(2)/(225)`
B. `(7)/(900)`
C. `(1)/(800)`
D. (number of prime numbers between 10 and 32)/900.

Answer

To solve the problem of finding the probability that a randomly chosen 3-digit number has exactly 3 factors, we can follow these steps:

### Step 1: Understand the Problem
We need to find the probability that a randomly chosen 3-digit number has exactly 3 factors. A number has exactly 3 factors if and only if it is a square of a prime number.

### Step 2: Identify the Range of 3-Digit Numbers
The range of 3-digit numbers is from 100 to 999. We can calculate the total number of 3-digit numbers:
- Total 3-digit numbers = 999 - 100 + 1 = 900.

### Step 3: Identify Prime Numbers Whose Squares are 3-Digit Numbers
Next, we need to find the prime numbers whose squares fall within the range of 3-digit numbers (100 to 999).

- The smallest prime number whose square is a 3-digit number is 11 (since \(11^2 = 121\)).
- The largest prime number whose square is still a 3-digit number is 31 (since \(31^2 = 961\)).

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

### Step 5: Count the Prime Numbers
Now, we count the prime numbers identified:
- There are 7 prime numbers: 11, 13, 17, 19, 23, 29, and 31.

### Step 6: Calculate the Probability
The probability \(P\) that a randomly chosen 3-digit number has exactly 3 factors is given by the formula:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{7}{900}. \]

### Final Answer
Thus, the probability that a randomly chosen 3-digit number has exactly 3 factors is \(\frac{7}{900}\).

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