To prove that for every prime $p > 5$, either $p^2 - 1$ or $p^2 + 1$ is divisible by 10, we can proceed by considering the properties of prime numbers modulo 10.

Step 1: Consider the possible values of $p$ modulo 10

Since $p$ is a prime greater than 5, the possible values of $p$ modulo 10 are: $p \equiv 1, 3, 7, 9 \pmod{10}$ This is because primes $p$ cannot be even, so $p$ is not congruent to 0, 2, 4, 6, or 8 modulo 10. Also, $p$ cannot be 5 modulo 10 since we are considering primes greater than 5.

Step 2: Calculate $p^2$ modulo 10

For each of the possible values of $p$ modulo 10, we calculate $p^2$ modulo 10:

1. If $p \equiv 1 \pmod{10}$: $p^2 \equiv 1^2 = 1 \pmod{10}$ Then, $p^2 - 1 \equiv 1 - 1 = 0 \pmod{10}$ So, $p^2 - 1$ is divisible by 10.

2. If $p \equiv 3 \pmod{10}$: $p^2 \equiv 3^2 = 9 \pmod{10}$ Then, $p^2 + 1 \equiv 9 + 1 = 10 \equiv 0 \pmod{10}$ So, $p^2 + 1$ is divisible by 10.

3. If $p \equiv 7 \pmod{10}$: $p^2 \equiv 7^2 = 49 \equiv 9 \pmod{10}$ Then, $p^2 + 1 \equiv 9 + 1 = 10 \equiv 0 \pmod{10}$ So, $p^2 + 1$ is divisible by 10.

4. If $p \equiv 9 \pmod{10}$: $p^2 \equiv 9^2 = 81 \equiv 1 \pmod{10}$ Then, $p^2 - 1 \equiv 1 - 1 = 0 \pmod{10}$ So, $p^2 - 1$ is divisible by 10.

Conclusion

In all cases, either $p^2 - 1$ or $p^2 + 1$ is divisible by 10 for primes $p > 5$.

Therefore, we have proven that for every prime $p > 5$, either $p^2 - 1$ or $p^2 + 1$ is divisible by 10.

Do you want any more details or have any questions?

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Related Questions:

1. How does modulo arithmetic help in proving divisibility properties?
2. Can we extend this proof to all odd numbers greater than 5, not just primes?
3. What are the properties of primes modulo small numbers like 10 or 5?
4. How do different modulo systems (e.g., mod 3, mod 4) affect similar proofs?
5. What is the significance of quadratic residues in number theory?
6. How does this property relate to the properties of twin primes?
7. Can we derive any similar divisibility rules using mod 12 or mod 15?
8. Is there a generalized form of this proof for primes of the form $p^2 + k$?

Tip: When working with modulo arithmetic, it is often helpful to test your hypotheses on small cases before generalizing, as it can provide insights into the underlying patterns.