The answer to exercise marked [BB] can be found in the Back of the Book.
(a) Given integers a, b, c, d, x, and a prime p, suppose (ax + b)(cx +d)≡0 (mod p). Prove that ax + b = 0 (mod p)or cx + d = 0 (mod p).
(b) Find all integers x, 0 ≤ x < n, that satisfy each of the following congruences. If no x exists, explain why not.
i. [BB] x2 = 4 (mod n), n = 13
ii. (2x + 1) (3x + 4) = 0 (mod n), n = 17
iii. 3x2 + 14x − 5 ≡ 0 (mod n), n = 97
iv. [BB] x2 ≡ 2(mod n), n = 6
v. x2 ≡ −2 (mod n), n = 6
vi. 4x2 + 3x + 7 ≡ 0 (mod n), n = 5
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