The number 4 is the smallest positive 
integer that has exactly three factors: 
1, 2, and 4. If k is the next-highest integer 
that also has exactly three factors, what is 
the sum of the three factors of k?

To solve the problem, we need to find the next-highest integer \( k \) that has exactly three factors, just like the number 4.

### Step 1: Understanding numbers with exactly three factors
If a number has exactly three factors, it means it must be of the form \( n = p^2 \), where \( p \) is a prime number. This is because:

- The total number of factors of a number \( n \) can be found by the formula:
  \[
  \text{Number of factors of } n = (e_1 + 1)(e_2 + 1)\dots(e_k + 1)
  \]
  where \( n = p_1^{e_1} \cdot p_2^{e_2} \dots p_k^{e_k} \) is the prime factorization of \( n \).
  
- For \( n \) to have exactly 3 factors, it must be the square of a prime. If \( n = p^2 \), its factors are \( 1, p, p^2 \), which makes 3 factors.

### Step 2: Finding the next number after 4
The number 4 is \( 2^2 \), and its factors are \( 1, 2, 4 \). We need to find the next prime number after 2 and compute the square of that prime.

- The next prime number after 2 is 3.
- The square of 3 is \( 3^2 = 9 \).

### Step 3: Finding the sum of the factors of 9
The factors of 9 are \( 1, 3, 9 \). The sum of these factors is:
\[
1 + 3 + 9 = 13
\]

Thus, the sum of the three factors of \( k = 9 \) is \( \boxed{13} \).

Would you like further details or have any questions?

Here are 5 related questions for further practice:
1. What are the factors of 25, and how many factors does it have?
2. What is the smallest number with exactly 4 factors?
3. Find the sum of the factors of \( p^3 \), where \( p \) is a prime.
4. What are the factors of 49, and how many factors does it have?
5. How many factors does the number 36 have?

**Tip:** The number of factors of a number can be determined from its prime factorization, and understanding this concept is essential for various problems in number theory.

The number 4 is the smallest positive integer that has exactly three factors: 1, 2, and 4. If k is the next-highest integer that also has exactly three factors, what is the sum of the three factors of k?

Discover how to find the next number after 4 with exactly three factors and calculate the sum of those factors. This problem involves number theory, prime factorization, and the theorem that numbers with three divisors are squares of primes. Suitable for Grades 6-8.

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