Question

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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.

Since, by assumption, w divides x, then x = kw for some integer k. Since, by assumption, y divides z, then z = ky for some integer k. Plug in the expression kw for x and ky for z in the expression xz to get

xz=(kw)(ky)=(k2)(wy)

Since k is an integer, then k² is also an integer. Since xz equals wy times an integer, then wy divides xz. ■

(b)Proof.

Since, by assumption, w divides x, then x = kw for some integer k. Since, by assumption, y divides z, then z = jy for some integer j. Let m be an integer such that xz = m·wy. Since xz equals wy times an integer, then wy divides xz. ■

(c)Proof.

Since, by assumption, w divides x, then x = kw for some integer k. Since, by assumption, y divides z, then z = jy for some integer j. Plug in the expression kw for x and jy for z in the expression xz to get

xz=(kw)(jy)

Since xz equals wy times an integer, then wy divides xz. ■

(d)Proof.

Since w divides x, then x = kw. Since, by assumption, y divides z, then z = jy. Plug in the expression kw for x and jy for z in the expression xz to get

xz=(kw)(jy)=(kj)(wy)

Since k and j are integers, then kj is also an integer. Since xz equals wy times an integer, then wy divides xz. ■

Answer 1