Q1

Solve the congruence


Q2

For what value of a does the congruence

have exactly one solution?

Q3

Show that if

is prime, then

Hint: order of an element modulo a prime p divides p-1

Q4

How many elements of order 6 are there modulo 13? List them all.
How many elements of order 5 are there modulo 13?

Q5

How many polynomials

of degree k with coefficients 0 and 1 don't have any roots modulo 2?

How many have exactly two roots?

Can such a polynomial have more than two roots?