(a) For integers a, b, q, r, prove that if a = bq+r then (a, b) = (b, r). (b) Determine (1000, 2025) by using the Euclidean algorithm. Then express the answer as a linear combination of 1000 and 2025. Show all your working. (c) Suppose that G = (a), a e, and a = e. Construct a Cayley table for the group (G,.). (d) Determine the right cosets of ((1 2), [1]) € S₂ x Z₂. (e) Find the subgroups of Z₂4.

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