I realise there are a lot of parts to this question, but any help with any of them would be much appreciated!
Let b1, b2, n1 and n2 be integers with gcd(n1, n2) = 1. Consider the congruences
x ≡ b1 (mod n1) (i)
x ≡ b2 (mod n2) (ii)
(a) Explain why the congruence n1x ≡ 1 (mod n2) has a solution.
(b) Letx = y1 be a solution to the congruence n1x ≡ 1 (mod n2) and x = y2 be a solution to n2x ≡ 1 (mod n1). Show that:(d) Find integers y1 and y2 such that 14y1 ≡ 1 (mod 15) and 15y2 ≡ 1 (mod 14).
x = b1n2y2 + b2n1y1
(c) Show that if x = s and x = t are solutions to both (i) and (ii) then
s ≡ t (mod n1n2).
(e) Use the answer to the previous parts of the question to find an integers such that:
x= s is a solution to both the following congruences simultaneously.
x≡ 2 (mod 14)
x ≡ 5 (mod 15).