Proving Exactly One Out of Three Consecutive Integers is Divisible by 3

In this article, we delve into the fascinating world of number theory by proving that exactly one of the three consecutive integers ( n ), ( n-1 ), and ( n 1 ) is divisible by 3, where ( n ) is any positive integer. This proof will be an exploration of the properties of modular arithmetic, a fundamental concept in number theory.

Introduction to Modular Arithmetic and Divisibility

Modular arithmetic, also known as the arithmetic of remainders, is a system in which numbers "wrap around" upon reaching a certain value—the modulus. In this context, we are particularly interested in divisibility by 3.

Divisibility by 3 Using Modular Arithmetic

When we divide an integer ( n ) by 3, the only possible remainders are 0, 1, and 2. These remainders help us understand the divisibility of ( n ), ( n-1 ), and ( n 1 ) by 3.

Step 1: Analyzing the Remainders

If ( n equiv 0 pmod{3} ), then ( n ) is divisible by 3.

If ( n equiv 1 pmod{3} ), then ( n ) is not divisible by 3, but ( n-1 ) is.

If ( n equiv 2 pmod{3} ), then ( n ) is not divisible by 3, but ( n 1 ) is.

Step 2: Case Analysis

Let's analyze each case in detail:

Case 1: ( n equiv 0 pmod{3} )

1. ( n equiv 0 pmod{3} ), thus ( n ) is divisible by 3.

2. ( n-1 equiv 1 pmod{3} ), hence ( n-1 ) is not divisible by 3.

3. ( n 1 equiv 2 pmod{3} ), hence ( n 1 ) is not divisible by 3.

Conclusion: Only ( n ) is divisible by 3.

Case 2: ( n equiv 1 pmod{3} )

1. ( n equiv 1 pmod{3} ), thus ( n ) is not divisible by 3.

2. ( n-1 equiv 0 pmod{3} ), hence ( n-1 ) is divisible by 3.

3. ( n 1 equiv 2 pmod{3} ), hence ( n 1 ) is not divisible by 3.

Conclusion: Only ( n-1 ) is divisible by 3.

Case 3: ( n equiv 2 pmod{3} )

1. ( n equiv 2 pmod{3} ), thus ( n ) is not divisible by 3.

2. ( n-1 equiv 1 pmod{3} ), hence ( n-1 ) is not divisible by 3.

3. ( n 1 equiv 0 pmod{3} ), hence ( n 1 ) is divisible by 3.

Conclusion: Only ( n 1 ) is divisible by 3.

Summary and Final Conclusion

By examining all three cases, we have shown that for any positive integer ( n ), exactly one of the three consecutive integers ( n ), ( n-1 ), and ( n 1 ) is divisible by 3. This result is a powerful statement about the structure of integers and their divisibility properties.

Alternative Approach Using the Product of Consecutive Integers

We can also prove this by considering the product of three consecutive integers ( n ), ( n-1 ), and ( n 1 ). The product of any three consecutive integers is divisible by 3! (which is 6). Since 3 is a factor of 6, exactly one of these three integers is divisible by 3.

Therefore, we conclude: For any positive integer ( n ), exactly one of the integers ( n ), ( n-1 ), and ( n 1 ) is divisible by 3.