Math 2510 Spring 2010 -- Homework Problem Set #3

Due: Friday, February 19th
  1. Use induction to show that \( 2^n < n! \) for all integers \( n \geq 4 \).

  2. Prove that 8 is a factor of \( 9^n - 1 \) for all natural numbers \( n \).

  3. Prove that \( n^2+n \) is always even [when \( n \) is an integer].

  4. Prove Theorem 20 (page 80): If no natural number \( m \) such that \( 1 < m \leq \sqrt{p} \) divides \( p \), then \( p \) is prime.

  5. Prove Theorem 34 (page 82): The sum of any three consecutive integers is divisible by 3.