SUMMARY
The discussion centers on proving that 6 divides the expression \( n^3 - n \) for all integers \( n \). The expression can be factored as \( n(n-1)(n+1) \), which represents the product of three consecutive integers. It is established that among any three consecutive integers, at least one is even (ensuring divisibility by 2) and at least one is divisible by 3, thus confirming that \( n^3 - n \) is divisible by 6 for all integers \( n \).
PREREQUISITES
- Understanding of integer properties and divisibility rules
- Familiarity with polynomial factorization
- Basic knowledge of modular arithmetic
- Concept of consecutive integers
NEXT STEPS
- Study the properties of divisibility in number theory
- Learn about polynomial identities and their proofs
- Explore modular arithmetic, specifically focusing on congruences
- Investigate the concept of factorials and their applications in combinatorics
USEFUL FOR
This discussion is beneficial for students studying number theory, mathematics educators, and anyone interested in proofs involving divisibility and polynomial expressions.