Odd Squares and Divisibility by 8

This problem explores a classic number theory result: the square of any odd integer, when reduced by 1, is always a multiple of 8. For HSC Mathematics Extension 1 students, it’s a great exercise in algebraic manipulation and logical reasoning, reinforcing how to represent odd numbers and use properties of consecutive integers. By working through this proof, you’ll gain confidence in constructing direct proofs and spotting factor tricks that simplify divisibility arguments.

Problem Statement

Prove that if $a$ is any odd integer, then $a^2 - 1$ is divisible by 8.

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Hints

Attempt the proof independently first. Focus on the key theorem, algebraic transformation, or contradiction setup that links the hypothesis to the target conclusion.

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Solutions

Direct Proof

Step 1: Express $a$ as an odd integer

To start the proof, we express the odd integer $a$ using its definition. Since $a$ is odd, we can write:

$$a = 2k + 1$$

for some integer $k$.

Step 2: Expand $a^2 - 1$

Now we substitute this expression into $a^2 - 1$ and simplify.

$$\begin{aligned} a^2 - 1 &= (2k + 1)^2 - 1 \\ &= 4k^2 + 4k + 1 - 1 \\ &= 4k^2 + 4k \\ &= 4k(k + 1) \end{aligned}$$

Step 3: Analyze $k(k+1)$

Note that $k$ and $(k+1)$ are consecutive integers.

Therefore, one of them must be even, which means their product $k(k+1)$ is divisible by 2.

Write $k(k+1) = 2m$ for some integer $m$.

Step 4: Substitute and conclude

Finally, we substitute this factorization back into our expression for $a^2-1$.

$$\begin{aligned} a^2 - 1 &= 4 \cdot k(k+1) \\ &= 4 \cdot 2m \\ &= 8m \end{aligned}$$

Since $a^2 - 1 = 8m$, we conclude that $a^2 - 1$ is divisible by 8.

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Takeaways

• Odd Integer Form: Any odd integer can be written as $2k+1$ for some integer $k$
• Consecutive Integer Property: Product of consecutive integers $k(k+1)$ is always even (one must be even)
• Factor Extraction: From $4k(k+1) = 4 \cdot 2m = 8m$, directly see divisibility by 8
• Alternative View: Can also factor $a^2-1 = (a-1)(a+1)$, both even for odd $a$, with one divisible by 4

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Further Readings

If you found this proof interesting, be sure to check out these relevant HSC booklets to sharpen your reasoning skills: HSC Proofs, HSC Last Resorts, HSC Mechanics