MATH 235 01: Basic Algebra Fall 2002

Assignment 2

1) Prove that gcd(n, n + 1) = 1 for every integer n.

2) Find the smallest positive integer in the set {6u + 15v | u, v ∈ Z}.

3) Prove that if a is odd then 4 | (a^{2} − 1).

4) Prove that if a is odd and b is even then gcd(a, b) = gcd(a + b, a − b).

5) Prove that if 3 | (a^{2} + b^{2}) then 3 | a and 3 | b.

6) Prove that \sqrt{30} is irrational (i.e., \sqrt{30} cannot be presented in the form \frac{m}{n} where m, n ∈ Z).

7) Show that 2^{11} − 1 is not a prime number.

8) Find all positive integers n such that n, n + 2, and n + 4 are all prime.