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.