Define and explain the significance of prime numbers in number theory and their relevance to cryptography.

Prime numbers play a fundamental role in number theory and have significant relevance to cryptography. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples of prime numbers include 2, 3, 5, 7, 11, and so on. The significance of prime numbers in number theory stems from their unique properties and characteristics. One key property is that every integer greater than 1 can be expressed as a product of prime numbers in a unique way, known as the prime factorization theorem. This theorem forms the foundation of many number theoretic algorithms and computations. In the context of cryptography, prime numbers serve as the building blocks for various encryption schemes and cryptographic algorithms. The security of many cryptographic systems relies on the difficulty of certain mathematical problems involving prime numbers. Two specific ap....