The probability that a randomly chosen 3-digit number has exactly 3 factors is k/900 then k is______.

Answer

To solve the problem of finding the value of \( k \) such that the probability that a randomly chosen 3-digit number has exactly 3 factors is \( \frac{k}{900} \), 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 the square of a prime number.

### Step 2: Identify the Range of 3-Digit Numbers
The smallest 3-digit number is 100 and the largest is 999. Therefore, our sample space consists of all integers from 100 to 999.

### Step 3: Count the Total Number of 3-Digit Numbers
The total number of 3-digit numbers can be calculated as:
\[ 999 - 100 + 1 = 900 \]
So, the sample space size is 900.

### Step 4: Identify Prime Numbers whose Squares are 3-Digit Numbers
Next, we need to find the prime numbers whose squares fall within the range of 100 to 999. We will find the prime numbers \( p \) such that:
\[ 10 \leq p \leq 31 \]
This is because \( 10^2 = 100 \) and \( 31^2 = 961 \) are the bounds for our 3-digit numbers.

### Step 5: List the Prime Numbers in the Range
The prime numbers between 10 and 31 are:
- 11
- 13
- 17
- 19
- 23
- 29
- 31

### Step 6: Calculate the Squares of These Prime Numbers
Now, we calculate the squares of these prime numbers:
- \( 11^2 = 121 \)
- \( 13^2 = 169 \)
- \( 17^2 = 289 \)
- \( 19^2 = 361 \)
- \( 23^2 = 529 \)
- \( 29^2 = 841 \)
- \( 31^2 = 961 \)

All of these squares are indeed 3-digit numbers.

### Step 7: Count the Favorable Outcomes
We have identified 7 prime numbers whose squares are 3-digit numbers. Thus, the number of favorable outcomes is 7.

### Step 8: Calculate the Probability
The probability \( P \) that a randomly chosen 3-digit number has exactly 3 factors is given by:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{7}{900} \]

### Step 9: Identify \( k \)
From the probability \( P = \frac{k}{900} \), we can see that \( k = 7 \).

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{7} \]