Step 1: Understanding the Question:
The question asks for the Highest Common Factor (HCF) of two numbers, \(m\) and \(n\), given that both are prime numbers.
Step 2: Key Formula or Approach:
The solution relies on the fundamental definitions of a prime number and the HCF.
Prime Number: A natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself.
HCF (Highest Common Factor): The largest positive integer that divides two or more numbers without leaving a remainder.
Step 3: Detailed Explanation:
Let's analyze the factors of the prime numbers \(m\) and \(n\).
By definition, the factors of the prime number \(m\) are \(\{1, m\}\).
Similarly, the factors of the prime number \(n\) are \(\{1, n\}\).
We consider two possible cases:
Case 1: $m$ and $n$ are different prime numbers (e.g., $m=3, n=5$).
The factors of \(m\) are \(\{1, m\}\).
The factors of \(n\) are \(\{1, n\}\).
The only factor that is common to both lists is 1. Therefore, the HCF is 1.
Case 2: $m$ and $n$ are the same prime number (i.e., $m=n$).
The factors of \(m\) are \(\{1, m\}\).
The factors of \(n\) (which is also \(m\)) are \(\{1, m\}\).
The common factors are 1 and \(m\). The highest common factor is \(m\).
Unless specified otherwise, questions like this usually imply that the numbers are distinct.
Step 4: Final Answer:
Assuming \(m\) and \(n\) are distinct prime numbers, their HCF is 1.
\[
\boxed{1}
\]