Prime numbers are a fundamental concept in mathematics with various applications in fields such as cryptography, computer science, and number theory. To determine whether the number 67 is a prime number, we need to understand the definition of a prime number and apply some basic rules for identifying prime numbers.

**What is a Prime Number?**

A **prime number** is a natural number greater than 1 that has no positive divisors other than 1 and itself. In other words, a prime number is a number that cannot be formed by multiplying two smaller natural numbers. For example, 2, 3, 5, 7, 11, and 13 are all prime numbers because they are only divisible by 1 and themselves.

**Is 67 a Prime Number?**

To determine whether 67 is a prime number, we need to check if it has any divisors other than 1 and 67. In the case of 67, the only numbers that divide evenly into 67 are 1 and 67 itself. Since 67 does not have any other divisors, it meets the definition of a prime number.

Therefore, **67 is a prime number**.

**Properties of Prime Numbers**

Prime numbers have several unique properties that make them distinct from other numbers. Some of the key properties of prime numbers include:

**Multiplicative Identity:**Every prime number multiplied by 1 gives the prime number itself.**Infinitude:**There is an infinite number of prime numbers.**Unique Factorization:**Every integer greater than 1 can be uniquely factored into a product of prime numbers.

**Prime Number Tests**

There are several methods to test for prime numbers, including:

**Trial Division:**Checking all possible divisors up to the square root of the number.**Sieve of Eratosthenes:**A method to generate all prime numbers up to a specified limit.**Fermat’s Little Theorem:**A probabilistic test for primality based on modular arithmetic.

**FAQs about Prime Numbers**

**1. How many prime numbers are there?**

There are infinitely many prime numbers. The list of prime numbers extends to infinity without any largest prime number.

**2. What is the smallest prime number?**

The smallest prime number is 2, as it is only divisible by 1 and 2 itself.

**3. Are negative numbers considered prime?**

By convention, prime numbers are defined as positive integers greater than 1. Negative numbers are not considered prime numbers.

**4. Can 1 be considered a prime number?**

No, 1 is not considered a prime number because it only has one positive divisor (1 itself). Prime numbers, by definition, must have exactly two positive divisors.

**5. How are prime numbers used in cryptography?**

Prime numbers play a crucial role in modern cryptography, particularly in encryption algorithms such as RSA. The security of these algorithms is based on the difficulty of factoring large composite numbers into their prime factors.

Understanding prime numbers and their properties is essential in various mathematical and scientific disciplines. By recognizing the unique characteristics of prime numbers, we can appreciate their significance and applications in different contexts.