Consider the following \"proof\" and \"counterexample\" to the conjecture \"If x

Question

Consider the following "proof" and "counterexample" to the conjecture

"If x,y,z are natrual numbers such that xz divides yz, the x divides y."

Is the "proof" correct? If not, why? Is the "counterexample" correct? If not, why?

"Proof:" Suppose x,y,z are natrual numbers such that xz divides yz but x does not divides y. Then y divides x, so there is an integer p such that x=py. Thus xz = (py)z = p(yz). Therefore yz divides xz, a coutradiction, since xz divides yz.

"counterexample:" Let x=4, y=6, and z=16. Then xz divides yz, but x does not divide y.

Explanation / Answer

The given proof is incorrect

It is given xz divides yz => yz/xz = p

Hence we get y/x = p

Therefore, the statement is correct and not incorrect

The counter example to proof the statement is also wrong

Given x=4, y=6, and z=16

xz = 4 * 16 = 64, yz = 6 * 16 = 96

xz divides yz is not true since 64 doesn't divide 96

Hence the counter example to disprove the statement is also wrong

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