Let $k$ be a positive integer. Consider $k$ not necessarily distinct prime numbers such that their product is ten times their sum. What are these primes and what is the value of $k$?
1. Let the primes be \( p_1, p_2, \ldots, p_k \). We are given that:
   \[
   10 \sum_{i=1}^k p_i = \prod_{i=1}^k p_i
   \]
   This equation states that the product of the primes is ten times their sum.

2. Since the product of the primes is ten times their sum, and considering the properties of prime numbers, it is reasonable to assume that two of the primes must be 2 and 5. This is because the product needs to be divisible by 10, which is the product of 2 and 5. Assume \( p_1 = 2 \) and \( p_2 = 5 \).

3. Substituting \( p_1 = 2 \) and \( p_2 = 5 \) into the equation, we get:
   \[
   10 \left(2 + 5 + \sum_{i=3}^k p_i \right) = 2 \cdot 5 \cdot \prod_{i=3}^k p_i
   \]
   Simplifying, we have:
   \[
   10 \left(7 + \sum_{i=3}^k p_i \right) = 10 \prod_{i=3}^k p_i
   \]
   Dividing both sides by 10, we get:
   \[
   7 + \sum_{i=3}^k p_i = \prod_{i=3}^k p_i
   \]

4. Let \( \sum_{i=3}^k p_i = x \) and \( \prod_{i=3}^k p_i = y \). Then we have:
   \[
   7 + x = y
   \]

5. By the AM-GM inequality, we know that:
   \[
   \frac{x}{k-2} \ge \sqrt[k-2]{y}
   \]
   which implies:
   \[
   \left( \frac{x}{k-2} \right)^{k-2} \ge y
   \]

6. Substituting \( y = 7 + x \) into the inequality, we get:
   \[
   \left( \frac{x}{k-2} \right)^{k-2} \le 7 + x
   \]

7. To find the value of \( k \), we need to solve this inequality. It is not hard to see that this implies \( k = 4 \). If \( k = 4 \), then we have:
   \[
   7 + p_3 + p_4 = p_3 p_4
   \]

8. Solving the equation \( 7 + p_3 + p_4 = p_3 p_4 \), we get:
   \[
   p_3 p_4 - p_3 - p_4 = 7
   \]
   Adding 1 to both sides, we get:
   \[
   (p_3 - 1)(p_4 - 1) = 8
   \]

9. The pairs of integers whose product is 8 are (2, 4) and (4, 2). Since \( p_3 \) and \( p_4 \) are primes, the only possible pairs are (3, 5) and (5, 3).

10. Therefore, the primes are \( 2, 3, 5, 5 \).

The final answer is \( \boxed{ (2, 3, 5, 5) } \)