`f:xto` highest prime factor of x.
A set of positive integers is called S. What can be said about these integers if (f)(S)=S?

Answer

To solve the problem, we need to analyze the function \( f(x) \) defined as the highest prime factor of \( x \) and the condition \( f(S) = S \), where \( S \) is a set of positive integers.

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( f(x) \) takes a positive integer \( x \) and returns its highest prime factor. For example:
- \( f(1) = 1 \) (since 1 has no prime factors)
- \( f(2) = 2 \) (2 is prime)
- \( f(3) = 3 \) (3 is prime)
- \( f(4) = 2 \) (highest prime factor of 4 is 2)
- \( f(5) = 5 \) (5 is prime)
- \( f(6) = 3 \) (highest prime factor of 6 is 3)
- \( f(7) = 7 \) (7 is prime)

2. **Condition \( f(S) = S \)**:
The condition \( f(S) = S \) means that applying the function \( f \) to every element in the set \( S \) must yield the same set \( S \). This implies that for every integer \( x \) in \( S \), \( f(x) \) must also be an element of \( S \).

3. **Analyzing Elements of \( S \)**:
- If \( x \) is a prime number, then \( f(x) = x \). Thus, all prime numbers will satisfy the condition \( f(x) = x \).
- If \( x \) is a composite number, \( f(x) \) will be the highest prime factor of \( x \), which is less than \( x \) itself. Therefore, composite numbers cannot satisfy \( f(x) = x \) because \( f(x) \) would not equal \( x \).

4. **Conclusion about Set \( S \)**:
Since \( f(x) = x \) holds true only for prime numbers, the set \( S \) must consist solely of prime numbers. Therefore, we conclude that:
\[ S = \{ \text{all prime numbers} \} \]

### Final Answer:
The set \( S \) consists of all prime numbers.

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