Prove that among any n+2 positive integers, there exist two positive integers whose difference is divisible by 2n or whose sum is divisible by 2n
Proving a Divisibility Property Using the Pigeonhole Principle
In number theory, the Pigeonhole Principle is a fundamental concept that helps us establish the existence of certain relationships between objects. In this article, we will explore a specific application of this principle to prove a divisibility property among positive integers.
The problem statement claims that among any n+2 positive integers, there exist two positive integers whose difference is divisible by 2n or whose sum is divisible by 2n. To tackle this problem, we will use the Pigeonhole Principle, which states that if n items are put into m containers, with n > m, then at least one container must contain more than one item.
Applying the Pigeonhole Principle
Let's consider n+2 positive integers. We want to show that there exist two numbers among these integers whose sum or difference is divisible by 2n. To do this, we will construct n+1 pigeonholes based on the residue classes modulo 2n. The pigeonholes are defined as follows:
- {0, 2n}
- {1, 2n-1}
- {2, 2n-2}
- ...
- {n-1, n+1}
- {n}
Each of these pigeonholes represents a possible residue class modulo 2n. Since we have n+1 pigeonholes, we can put the n+2 positive integers into these pigeonholes.
The Pigeonhole Principle in Action
By the Pigeonhole Principle, since we have n+2 positive integers and only n+1 pigeonholes, at least two numbers must fall into the same pigeonhole. Let's call these two numbers x and y. Now, consider the sum and difference of x and y:
- Sum: x + y
- Difference: x - y
Since x and y fall into the same pigeonhole, their sum or difference must be divisible by 2n. This is because the residue classes modulo 2n are defined such that the sum or difference of two numbers in the same class is always divisible by 2n.
Conclusion
In this article, we used the Pigeonhole Principle to prove that among any n+2 positive integers, there exist two positive integers whose difference is divisible by 2n or whose sum is divisible by 2n. This result demonstrates the power of the Pigeonhole Principle in establishing relationships between objects in number theory.