Find the solutions of the congruence 15x^2 + 19x = 5 (mod 11). [Hint: Show the congruence is equivalent to the congruence 15x^2 + 19x + 6 = 0 (mod 11). Factor the left-hand side of the congruence; show that a solution of the quadratic congruence is a solution of on eof hte two different linear congruences.]



It was easy to follow the hint of adding 6 to both sides and factoring the left-hand side, but to solve the congruences, I didn't know how. The solution manual says that the inverse of 5 modulus 11 is 9, and multiplying both sides of 5x + 3 = 0 (mod 11) by 9 yeilds x + 27 = 0 (mod 11), so x = -27 = 6 (mod 11). Similarly, an inverse of 3 modulo 11 is 4, and we get x = -8 = 3 (mod 11). So the solution set is {3, 6} (and anything congruenct to these modulo 11).