MATH 319 Test 1

Instructions: Answer all the questions on the blank paper provided. Show all your work and be specific with your reasons when justifying a statement. Make sure that your name is on each sheet you hand in.

1. Given that 2 is a primitive root mod 19, answer each of the following questions:

a) What is = ?

b) How many primitive roots does 19 have?

c) Find two other primitive roots.

d) Find (if possible) an integer k such that . If it is not possible, why not?

e) Find (if possible) an integer k such that If it is not possible, why not?

f) How many integers in a reduced residue system mod 19 satisfy the congruence ?

2. Prove: Suppose that d and m are positive integers and . If then .

3. Find a solution to each of the following congruences or show that the congruence has no solution.

a)

b)

c)

4. Show that is not divisible by 11 for any integer n.

5. Given that and , what would you do next to determine if 52633 is prime or composite? Assume that you can not try to factor the number but that you can compute for various values of a and m.