Yes, every integer greater than 1 is divisible by at least one prime number.
Explanation
The statement that every integer greater than 1 is divisible by some prime number is a fundamental concept in number theory. Here's a breakdown of why this is true, drawing from the provided reference:
The Basis of the Argument
The provided reference states: "Every integer greater than 1 is divisible by a prime number." It also provides a proof outline:
- Base Case: The statement is true for the integer 2, since 2 is itself a prime number and divisible by itself (2 | 2).
- Inductive Step: Suppose the statement is true for all positive integers greater than 1 and less than n. The proof then considers the integer n. If n is prime, it is trivially divisible by itself, which is a prime. If n is composite (not prime), it must have at least one divisor other than 1 and itself. This divisor must also be an integer, and since it is less than n, the inductive hypothesis states it must be divisible by a prime. Thus, n itself will be divisible by the same prime.
What Does This Mean?
This concept can be broken down as follows:
- Prime Numbers: Prime numbers are whole numbers greater than 1 that have only two divisors: 1 and themselves. Examples include 2, 3, 5, 7, 11, etc.
- Divisibility: When we say a number is "divisible" by another, it means that the division results in a whole number (without any remainder).
- Composite Numbers: Composite numbers are whole numbers greater than 1 that have more than two divisors. Examples include 4, 6, 8, 9, 10, etc.
- Integers less than 2: Integers 1, 0, and negative integers are not considered by the statement. 1 is neither prime nor composite and the concept of divisibility by prime numbers doesn't apply similarly. 0 is divisible by any non-zero integer. Finally, the divisibility of negative numbers by prime numbers is analogous to positive integers.
Examples
Here are a few examples illustrating the concept:
- 4: Is divisible by the prime number 2.
- 15: Is divisible by the prime numbers 3 and 5.
- 23: Is divisible by the prime number 23 (since 23 is prime).
- 100: Is divisible by the prime numbers 2 and 5.
Key Takeaways
- Every integer greater than 1 can be broken down into prime factors.
- The existence of prime divisors is a fundamental property of integers.
- This property is vital in various aspects of number theory, cryptography, and computer science.
Exception
- The number 1 is not included, because 1 is neither a prime nor a composite number, and it doesn't have prime factors.
Conclusion
Therefore, the answer to the question "Is every integer divisible by some prime number?" is yes, if we are talking about integers greater than 1, in agreement with the provided reference.
