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Exercise 2.3.2: Find the mistake in the proof - integer division. Theorem: If w, x, y, z are integers where w divides x and y divides z, then wy divides xz. For each "proof" of the theorem, explain where the proof uses invalid reasoning or skips essential steps. (a) Proof. i About Let w, x, y, z be integers such that w divides x and y divides z. Since, by assumption, w divides x, then x = kw for some integer k and w #0. Since, by assumption, y divides z, then z = ky for some integer k and y # 0. Plug in the expression kw for x and ky for z in the expression xz to get xz= (kw)(ky) = (k) (wy) Since k is an integer, then k is also an integer. Since w #0 and y #0, then wy # 0. Since xz equals wy times an integer and wy # 0, then wy divides xz. Proof. Let w, x, y, z be integers such that w divides x and y divides z. Since, by assumption, w divides x, then x = kw for some integer k and w0. Since, by assumption, y divides z, then z = jy for some integer j and y # 0. Since w 0 and y 0, then wy 0. Let m be an integer such that xz = m-wy. Since xz equals wy times an integer and wy #0, then wy divides xz.