February+7

Q1

Solve the congruence math x^2+x-1\equiv 0 \mod 11 math

Q2

For what value of a does the congruence math x^2+x+a\equiv 0 \mod 11 math have exactly one solution?

Q3

Show that if math 2^n-1 math is prime, then math n|2^n-2 math 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 math x^k+a_1x^{k-1}+\ldots+a_k math 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?