Math Proof : n(n⁴−1) Divisible by 30 for All n
Автор: The Mathematiker
Загружено: 2025-08-31
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In this video, we prove that the expression n × (n⁴ − 1) is divisible by 30 for every natural number n.
We start by noting that since 30 = 5 × 6 and 5 and 6 are coprime, it suffices to show divisibility by both 5 and 6.
While divisibility by 6 is straightforward, we explore the divisibility by 5 through the five residue classes modulo 5:
n = 5k
n = 5k + 1
n = 5k + 2
n = 5k + 3
n = 5k + 4
After analyzing each case, we conclude that n(n⁴−1) is divisible by both 5 and 6, and therefore by 30.
This proof combines number theory, modular arithmetic,
and logical reasoning. Perfect for students preparing for exams or anyone passionate about mathematics!
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