Direct Proof of Prime Divisibility in Products
Prove that if a prime number (p) divides the product (ab), then (p) must divide (a) or (p) must divide (b).
Contradiction in Divisibility: A Case Study
Let \(n\) be an integer greater than 1. Prove by contradiction that no integer \(k\) with \(1 < k \leq n\) divides \(n! + 1\).
Mixed Proof Methods for Divisibility Identities
Consider the following statement: For all integers \( n \geq 1 \), the expression \( n^3 – n \) is divisible by 6. (a) Prove by induction that for every integer \( n \geq 1 \), \( n^3 – n \) is divisible by 6. (b) Using a proof by contradiction, show that there does […]
Part a
Prove by induction that for every integer \( n \geq 1 \), the expression \( n^3 – n \) is divisible by 6.
Part b
Using a proof by contradiction, show that there does not exist an integer \( n \) for which \( n^3 – n \) is not divisible by 6.