Reference no: EM132659933

Question 1: Suppose m₁, m₂, ..., m_k are positive integers > 1, not necessarily pairwise relatively prime. Also suppose a₁, a₂, a_k ∈ Z. Discuss the solutions of the following set of linear congruence equations in details. You can assume the statement of Chinese Remainder Theorem if and when necessary.

x ≡ a₁(mod m₁), x≡a₂ (mod m₂), x≡a₃ (mod m₃), ...... x ≡ ak (mod mk).

Question 2: Suppose m and n are two positive integers such that (m, n) = 7. Suppose f (x) a₀ + a₁x + a₂x² + a₃x³ is a polynomial where a_i ∈ Z for all i = 0, 1, 2, 3 and at least 3 of a₀, a₁, a₂, a₃ are non-zero. Also given a₀ ≠ 1.

Will there be 10 distinctly different examples of f (x) such that (f (m), f (n)) = 1? Justify in detail.

Question 3: Someone incorrectly remembered Fermat's Little Theorem as saying that the congruence aⁿ⁺¹ a (mod n) holds for all a if n is a prime. Describe the set of integers n for which this property is in fact true.