All integers greater than 1 are divisible by at least one prime number.
Here's a breakdown:
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Prime Numbers: Prime numbers are whole numbers greater than 1 that have only two divisors: 1 and themselves (e.g., 2, 3, 5, 7, 11, etc.).
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Divisibility: A number is divisible by another number if the result of their division is a whole number (i.e., there's no remainder).
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The Fundamental Theorem of Arithmetic: This theorem states that every integer greater than 1 can be uniquely represented as a product of prime numbers (up to the order of the factors). This is also known as prime factorization.
Examples:
- 6 is divisible by 2 and 3 (both are prime). 6 = 2 x 3
- 15 is divisible by 3 and 5 (both are prime). 15 = 3 x 5
- 28 is divisible by 2 and 7 (both are prime). 28 = 2 x 2 x 7
- 37 is divisible by 37 (itself, which is prime).
- Even Numbers: All even numbers greater than 2 are divisible by the prime number 2.
Why is this important?
The concept of prime factorization and divisibility by prime numbers is fundamental to many areas of mathematics, including:
- Cryptography: Prime numbers are used extensively in encryption algorithms.
- Number Theory: The study of prime numbers is a major branch of number theory.
- Computer Science: Prime numbers are used in hashing algorithms and data structures.
Therefore, understanding which numbers are divisible by prime numbers is key to grasping many important mathematical and computational concepts.
