Prove that 5 divides 𝑛⁵ − 𝑛 when 𝑛 is a nonnegative integer | Discrete Math

In this video we prove by induction that 5 divides n^5 - n for all nonnegative integers. This problem comes from Kenneth Rosen’s Discrete Mathematics and Its Applications (7th Edition, Chapter 5.1, Question 33). Even if you are using a different edition, the exercise appears in most versions of the text and is a classic induction problem on divisibility.

We begin by carefully reviewing what integer division means and what it means to say that one integer divides another. We then establish the base case and check values of n=0, 1, 2 to illustrate the pattern. The inductive step is built using the induction hypothesis, where we assume k^5 - k is divisible by 5 and show that
(k+1)^5−(k+1) is also divisible by 5. To expand (k+1)^5, we introduce the binomial theorem as a powerful tool for handling higher-degree polynomials in induction proofs.

Through step-by-step algebra, we factor terms, apply the induction hypothesis, and demonstrate that the result is an integer multiple of 5. This completes the proof that 5 divides n^5 - n for n greater than 0.

This lecture is part of the Dogmathic Discrete Mathematics series. If you are studying proofs by induction, divisibility problems, or working through Rosen’s textbook, this walkthrough will help you understand both the mechanics and the reasoning behind the proof. Like and subscribe for more videos on discrete mathematics, set theory, algebra, and advanced problem solving.

Discrete Mathematics and Its Applications playlist:
   • Discrete Mathematics  

Induction Proofs playlist:
   • Induction Proofs  

How to Factor Polynomials:
   • How to Factor Polynomials | College Algebr...  

How to multiply polynomials:
   • Adding, Subtracting, Multiplying, and Divi...  

Chapters:
00:00 Introduction
01:00 What is Integer Division
02:50 Base Case P(0)
04:41 Inductive Step
05:10 Induction Hypothesis P(k)
06:16 We Want to Show P(k+1)
07:36 Algebra Steps
09:08 Binomial Theorem (an aside)
18:45 Algebra Steps Continued
23:54 QED and Thanks for Watching

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