Verified Answer :

  • Define a prime number as an integer greater than 1 with no divisors other than 1 and itself.
  • Verify that for any prime number \(p\), the factors are strictly 1 and \(p\), resulting in exactly two factors.
  • Exclude the number 1 as it has only one factor, and exclude composite numbers which have more than two factors.
  • Conclude that the correct choice is option 2: \(\boxed{\text{exactly two factors}}\).

Explanation

  1. Definition of Prime Numbers

A prime number is defined as a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. This is equivalent to saying that a prime number has exactly two distinct positive divisors: 1 and the number itself.

  1. Examining Factors of Small Integers

Let's examine some small numbers:

  • The number 1: Its only factor is 1. It is not considered a prime number because it does not have two distinct factors.
  • The number 2: Its factors are 1 and 2. It has exactly two factors, so it is prime.
  • The number 3: Its factors are 1 and 3. It has exactly two factors, so it is prime.
  • The number 4: Its factors are 1, 2, and 4. It has three factors, so it is a composite number, not a prime.

From these examples, it is clear that any prime number \(p\) has exactly the set of factors \(\{1, p\}\).

  1. Conclusion regarding the number of factors

Based on the definition and our examination, any prime number possesses exactly two distinct positive divisors. Therefore, the statement "exactly two factors" is the correct classification. The other options are incorrect: prime numbers cannot have fewer than two (as 1 is excluded) and they cannot have more than two (otherwise they would be composite).

Examples

Understanding prime numbers is fundamental in fields like cryptography. For example, modern encryption methods, such as RSA, rely on the fact that while it is easy to multiply two large prime numbers together, it is extremely difficult to reverse the process and factor the resulting composite number. By knowing that prime numbers have only two factors, we can build secure systems that protect online information and communication.

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