Reference no: EM132660153
Question 1: Suppose m1, m2, ..., mk are positive integers > 1, not necessarily pairwise relatively prime. Also suppose a1, a2, ak ∈ 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 ai ∈ 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.
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