Among any three consecutive integers, one must be divisible by 3 (since every third integer is a multiple of 3). - American Beagle Club
Title: Why Among Any Three Consecutive Integers, One Must Be Divisible by 3
Title: Why Among Any Three Consecutive Integers, One Must Be Divisible by 3
Understanding basic number properties can unlock powerful insights into mathematics and problem-solving. One fundamental and elegant fact is that among any three consecutive integers, exactly one must be divisible by 3. This simple rule reflects the structure of whole numbers and offers a gateway to deeper mathematical reasoning. In this article, we’ll explore why every set of three consecutive integers contains a multiple of 3, how this connection to divisibility works, and why this principle holds universally.
Understanding the Context
The Structure of Consecutive Integers
Three consecutive integers can be written in the general form:
- n
- n + 1
- n + 2
Regardless of the starting integer n, these three numbers fill a block of three digits on the number line with a clear pattern. Because every third integer is divisible by 3, this regular spacing guarantees one of these numbers lands precisely at a multiple.
Key Insights
The Role of Modulo 3 (Remainders)
One way to prove this is by examining what happens when any integer is divided by 3. Every integer leaves a remainder of 0, 1, or 2 when divided by 3—this is the foundation of division by 3 (also known as modulo 3 arithmetic). Among any three consecutive integers, their remainders when divided by 3 must fill the complete set {0, 1, 2} exactly once:
- If n leaves remainder 0 → n is divisible by 3
- If n leaves remainder 1 → then n + 2 leaves remainder 0
- If n leaves remainder 2 → then n + 1 leaves remainder 0
No matter where you start, one of the three numbers will have remainder 0, meaning it is divisible by 3.
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Question: A hydrologist models groundwater flow with vectors $\mathbf{a} = \begin{pmatrix} 2 \\ -3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 4 \end{pmatrix}$. Find the angle between these flow directions. Solution: The angle $\theta$ between $\mathbf{a}$ and $\mathbf{b}$ is given by $\cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|}$. Compute the dot product: $2(1) + (-3)(4) = 2 - 12 = -10$. Compute magnitudes: $\|\mathbf{a}\| = \sqrt{2^2 + (-3)^2} = \sqrt{13}$, $\|\mathbf{b}\| = \sqrt{1^2 + 4^2} = \sqrt{17}$. Thus, $\cos\theta = \frac{-10}{\sqrt{13}\sqrt{17}}$. Rationalizing, $\theta = \arccos\left(-\frac{10}{\sqrt{221}}\right)$. $\boxed{\arccos\left(-\dfrac{10}{\sqrt{221}}\right)}$ Question: If $\mathbf{u}, \mathbf{v}, \mathbf{w}$ are unit vectors in hydrology modeling, find the maximum value of $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$.Final Thoughts
Examples That Illustrate the Rule
Let’s verify with specific examples:
- 3, 4, 5: 3 is divisible by 3
- 7, 8, 9: 9 is divisible by 3
- 13, 14, 15: 15 is divisible by 3
- −2, −1, 0: 0 is divisible by 3
- 100, 101, 102: 102 is divisible by 3
Even with negative or large integers, the same logic applies. The pattern never fails.
Why This Matters Beyond Basic Math
This property is not just a numerical curiosity—it underpins many areas of mathematics, including:
- Number theory, where divisibility shapes how integers behave
- Computer science, in hashing algorithms and modulo-based indexing
- Cryptography, where modular arithmetic safeguards data
- Everyday problem-solving, helping simplify counting, scheduling, and partitioning
