Math 580
Review Exam #1

1. Prove: is irrational.
   

2. Prove: If a and b are relatively prime integers, and a | bc then a | b
   

3. Let ,
   Then find the value of gcd(n, m) and lcm(n, m).
   How many divisors does n have?
   

4. Prove: For any two positive integers a and b, gcd(a, b)lcm(a, b) = ab
   

5. Prove: Let d be the greatest common divisor of the positive integers a and b. Let x and y be integers such that ax + by = d. Then x and y are relatively prime.

6. Prove: for any integer a show that gcd(4a + 1, 9a + 2) = 1
   

7. Show Using congruences that for every odd positive integer x, .
   

8. What is the remainder when 2¹⁰⁴ is divided by 11?
   

9. Prove: If 2ⁿ - 1 is a prime, then n is a prime.
   

10. Prove: There are an infinite number of prime integers.
    

11. Prove: If x, y and z are integers with x² + y² = z², then not both of x and y are odd.

    
    

From Your Text

Page 25 #3, 4(d), 21

Page 32 #1, 2(c), 8, 9

Page 39 #1, 2(c), 3(d), 7, 8

Page 45 #3, 4, 5, 6, 10, 12

Page 51 #1, 5

Page 60 #9(a), 10, 20, 22

Page 70 #1, 2, 4, 5, 6, 8, 16

Selected Solutions