Proving Exactly One Number Among n, n1, and n2 is Divisible by 3 for Any Positive Integer n

In this article, we will prove that for any positive integer n, exactly one of the numbers n, n1, or n2 is divisible by 3. We will explore this through the lens of modular arithmetic and Euclidean division.

Introduction

The topic of divisibility is fundamental in number theory. Here, we focus on the special case where among the numbers n, n1, and n2, only one can be divisible by 3. This property is not only intriguing but also has applications in various areas of mathematics, including number theory and algorithm design.

Using Modular Arithmetic

Module arithmetic is a powerful tool in number theory. We start by considering the possible values of n modulo 3:

n equiv; 0 (mod 3)

n equiv; 1 (mod 3)

n equiv; 2 (mod 3)

Case 1: n equiv; 0 (mod 3)

If n equiv; 0 (mod 3), then n1 equiv; 1 (mod 3), n2 equiv; 2 (mod 3), and n3 equiv; 0 (mod 3). In this case, n3 is clearly divisible by 3.

Case 2: n equiv; 1 (mod 3)

If n equiv; 1 (mod 3), then n1 equiv; 2 (mod 3), n2 equiv; 0 (mod 3), and n3 equiv; 1 (mod 3). In this case, n2 is divisible by 3.

Case 3: n equiv; 2 (mod 3)

If n equiv; 2 (mod 3), then n1 equiv; 0 (mod 3), n2 equiv; 1 (mod 3), and n3 equiv; 2 (mod 3). In this case, n1 is divisible by 3.

From these three cases, we see that in each scenario exactly one of n1, n2, or n3 is divisible by 3. This completes the proof using modular arithmetic.

Using Euclidean Division Lemma

The Euclidean Division Lemma states that any integer n can be expressed as:

n 3m r, where r can be 0, 1, or 2.

Case 1: n 3m

If n 3m, then n1 3m - 1 and n2 3m 1. Neither n1 nor n2 is divisible by 3.

Case 2: n1 3m

If n1 3m, then n 3m 1 and n2 3m 2. Neither n nor n2 is divisible by 3.

Case 3: n2 3m

If n2 3m, then n 3m - 1 and n1 3m - 2. Neither n nor n1 is divisible by 3.

From these cases, we can conclude that one and only one out of n, n1, and n2 is divisible by 3.

Conclusion

The proof that exactly one of the numbers n, n1, and n2 is divisible by 3 for any positive integer n has been demonstrated using both modular arithmetic and Euclidean division lemma. This result is not only mathematically elegant but also provides a solid foundation for further explorations in number theory.