1. ## Finite Arithmetic

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) Let
x = y1 be a solution to the congruence n1x 1 (mod n2) and x = y2 be a solution to n2x 1 (mod n1). Show that:

x = b1n2y2 + b2n1y1

(c) Show that if x = s and x = t are solutions to both (i) and (ii) then
s t (mod n1n2).
(d) Find integers y1 and y2 such that 14y1 1 (mod 15) and 15y2 1 (mod 14).

(e) Use the answer to the previous parts of the question to find an integer
s such that:

x
= s is a solution to both the following congruences simultaneously.

x
2 (mod 14)

x 5 (mod 15).

2. You are supposed to post 1 question per thread. I have a hard time imagining you cannot do any one of those yourself. You should at least try, and then let us know where you get stuck exactly. This website is not for other people to do your homework for you.