There are just four natural number less than 100, with exactly 3 factors. One of them is 25. What are the other three? What do you notice about these four numbers?

Answer

To solve the problem, we need to find all natural numbers less than 100 that have exactly three factors. We already know that one of these numbers is 25. Let's go through the steps to find the other three numbers.

### Step 1: Understand the condition for having exactly three factors
A natural number has exactly three factors if it is the square of a prime number. This is because the factors of \( p^2 \) (where \( p \) is a prime) are \( 1, p, \) and \( p^2 \).

### Step 2: Identify prime numbers less than 10
Since we are looking for squares of prime numbers that are less than 100, we first list the prime numbers less than 10:
- 2
- 3
- 5
- 7

### Step 3: Calculate the squares of these prime numbers
Now, we will calculate the squares of these prime numbers:
- \( 2^2 = 4 \)
- \( 3^2 = 9 \)
- \( 5^2 = 25 \) (already given)
- \( 7^2 = 49 \)

### Step 4: List the squares that are less than 100
From the calculations, we have:
- \( 4 \)
- \( 9 \)
- \( 25 \) (given)
- \( 49 \)

### Step 5: Identify the three numbers
The three natural numbers less than 100 with exactly three factors, along with 25, are:
- 4
- 9
- 49

### Step 6: Conclusion
Thus, the four natural numbers less than 100 with exactly three factors are:
- 4
- 9
- 25
- 49

### Step 7: Notice the commonality
The commonality among these numbers is that they are all squares of prime numbers.

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