Explanation: The highest common factor (HCF), also known as the greatest common divisor (GCD), of two numbers is the largest number that divides both of them without leaving a remainder. When considering prime numbers, it is important to understand their fundamental property: a prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
Given this definition, let's consider two distinct prime numbers, say \( p \) and \( q \). By definition, \( p \) and \( q \) have no divisors other than 1 and themselves. Therefore, the only number that can divide both \( p \) and \( q \) is 1. This means that the HCF of \( p \) and \( q \) is 1.
Now, let's analyze the nature of the number 1. The number 1 is neither prime nor composite; it is a unique number. However, it is an odd number. This is because odd numbers are integers that are not divisible by 2. Since 1 is not divisible by 2, it is classified as an odd number.
Thus, the HCF of any two prime numbers is 1, which is an odd number. This explains why option C is the correct answer.
To further solidify this understanding, let's consider a few examples:
- The HCF of 2 and 3 is 1.
- The HCF of 5 and 7 is 1.
- The HCF of 11 and 13 is 1.
In each of these cases, the HCF is 1, which is an odd number. This pattern holds true for any pair of distinct prime numbers.
It is important to note that the HCF of two prime numbers cannot be a prime number other than 1, a composite number, or an even number. The only common factor between two distinct prime numbers is 1, which is an odd number.
In conclusion, the HCF of any two prime numbers is always an odd number, specifically 1. This concept is fundamental in number theory and is often tested in various competitive examinations in India, such as UPSC, SSC, Banking Exams, CAT, and MAT. Understanding this concept thoroughly will help in solving related problems efficiently.