Reference no: EM131070062

1. Find the least complete solution: x³ + x² + 1 ≡ 0 (mod 27).

2. Find the least complete solution:  x² - x + 7 ≡ 0 (mod 117).

3. Find the least complete solution to the congruence. Start by making a table of the least residues of 1², 2², ...., 8² (mod 17).

5x² + 11x - 2 ≡ 0 (mod 17)

4. Suppose p is an odd prime nor dividing ab. Show that z² ≡ ab (mod p) is solved exactly when both or neither of x² ≡ a (mod p) and y² ≡ b (mod p) are solvable.

5. Use the law of quadratic reciprocity to show that if p is an odd prime other than 5, then (5/p) = 1 if and only if the last digit of p is 1 or 9.

6. Show that if ab ≡ 1 (mod p), p an odd prime, then (a/p) = (b/p).