Let's break down the definition of a prime number to understand the correct option. The question asks for the characteristic that defines a prime number.

Understanding Prime Numbers

A prime number is a special type of positive integer. To be a prime number, a number must meet two specific conditions:

1. It must be an integer greater than 1.
2. It must have exactly two distinct positive divisors. These two divisors must be the number 1 and the number itself.

Let's look at some examples:

• The number 2: Its positive divisors are 1 and 2. There are exactly two divisors. So, 2 is a prime number.
• The number 3: Its positive divisors are 1 and 3. There are exactly two divisors. So, 3 is a prime number.
• The number 4: Its positive divisors are 1, 2, and 4. There are three divisors. So, 4 is not a prime number. It is a composite number.
• The number 5: Its positive divisors are 1 and 5. There are exactly two divisors. So, 5 is a prime number.
• The number 6: Its positive divisors are 1, 2, 3, and 6. There are four divisors. So, 6 is not a prime number. It is a composite number.

The number 1 is not considered a prime number because it only has one positive divisor (which is 1), not two distinct divisors.

Analyzing the Options for Prime Numbers

Now let's evaluate each option based on the definition of a prime number:

Option A: A prime number is not a positive integer.

This statement is incorrect. By definition, a prime number is a positive integer that is greater than 1. Examples like 2, 3, 5, 7 are all positive integers.

Option B: A prime number has no divisor at all.

This statement is incorrect. Every integer (except 0) has divisors. For instance, any positive integer is divisible by 1 and by itself. Prime numbers specifically have divisors.

Option C: A prime number has only 1 and itself as divisors.

This statement accurately describes the defining characteristic of a prime number. A positive integer greater than 1 is prime if its only positive divisors are 1 and the number itself. This means it has exactly two distinct positive divisors.

Option D: A prime number has more than two divisors.

This statement is incorrect. Numbers that have more than two positive divisors are called composite numbers. For example, 4 has divisors 1, 2, and 4 (three divisors); 6 has divisors 1, 2, 3, and 6 (four divisors). These are composite numbers, not prime numbers.

Conclusion

Based on the definition and analysis, the only option that correctly describes a prime number is that it has only 1 and itself as divisors.

Revision Table: Key Concepts

Additional Information on Prime Numbers and Divisors

Prime numbers are the building blocks of integers through multiplication. This concept is formalized in the Fundamental Theorem of Arithmetic, which states that every integer greater than 1 can be uniquely represented as a product of prime numbers (ignoring the order of the factors).

For example, \( 12 = 2 \times 2 \times 3 = 2^2 \times 3 \). Here, 2 and 3 are prime numbers.

Understanding divisors is crucial for identifying prime and composite numbers. The number of divisors a number has depends on its prime factorization. If a number \( N \) has a prime factorization \( N = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k} \), where \( p_i \) are distinct prime numbers and \( a_i \) are positive integers, then the total number of positive divisors of \( N \) is given by \( (a_1 + 1)(a_2 + 1)\cdots(a_k + 1) \). A prime number \( p \) has the factorization \( p^1 \), so its number of divisors is \( (1+1) = 2 \).