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Let a, b, and n be integers with n > 1 and (a, n) = d....

Let a, b, and n be integers with n > 1 and (a, n) = d. Then

(i)First prove that the equation a·x=b has solutions in n if and only if d|b.

(ii) Next, prove that each of u, u+n′, u+ 2n′, . . . , u+ (d−1)n′ is a solution. Here,u is any particular solution guaranteed by (i), and n′=n/d.

(iii) Show that the solutions listed above are distinct.

(iv) Let v be any solution. Prove that v=u+kn′ for some k ∈ with 0≤k

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