Using the hint that the square of odd integer is of the form 8k+1, prove that if a is an odd integer, then 24|a(a^2-1).

We are given that \( a \) is an odd integer, and we need to prove that \( 24 \) divides \( a(a^2 - 1) \).

### Step 1: Express \( a \) as an Odd Integer
Let \( a \) be an odd integer. Any odd integer can be written in the form \( a = 2n + 1 \), where \( n \) is an integer.

### Step 2: Calculate \( a^2 - 1 \)
Now, calculate \( a^2 - 1 \):

\[
a^2 = (2n + 1)^2 = 4n^2 + 4n + 1
\]
\[
a^2 - 1 = (4n^2 + 4n + 1) - 1 = 4n^2 + 4n = 4n(n + 1)
\]

### Step 3: Analyze \( a(a^2 - 1) \)
We need to compute \( a(a^2 - 1) \):

\[
a(a^2 - 1) = (2n + 1) \times 4n(n + 1)
\]

Thus,

\[
a(a^2 - 1) = 4(2n + 1)n(n + 1)
\]

### Step 4: Show Divisibility by 24
To show that \( 24 \) divides \( 4(2n + 1)n(n + 1) \), we need to prove that \( 6 \) divides \( (2n + 1)n(n + 1) \). Note that:

- \( n \) and \( n + 1 \) are consecutive integers, so one of them must be divisible by 2. Therefore, \( n(n + 1) \) is divisible by 2.
- Additionally, among any three consecutive integers, one of them must be divisible by 3. Since \( (2n + 1) \), \( n \), and \( n + 1 \) are three consecutive integers, one of them is divisible by 3.

Therefore, \( (2n + 1)n(n + 1) \) is divisible by both 2 and 3, implying that it is divisible by 6. Multiplying by 4 gives a factor of \( 24 \).

Thus, \( 24 \) divides \( a(a^2 - 1) \), completing the proof.

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Do you have any questions or would you like further details? Here are five related questions to explore:

1. Can you prove the same result for even integers?
2. How does the factorization of polynomials help in divisibility proofs?
3. What other forms do the squares of integers take besides \( 8k + 1 \) for odd integers?
4. Can you extend this proof to higher powers of odd integers, such as \( a^3 - 1 \)?
5. How does modular arithmetic apply in proving divisibility properties?

**Tip:** Always break down complex proofs into smaller, manageable steps to ensure clarity and accuracy.

Learn how to prove that if a is an odd integer, then 24 divides a(a^2 - 1). This detailed mathematical proof involves understanding the form of odd integers and their divisibility properties. Suitable for high school students studying number theory and divisibility proofs.

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