Data Sufficiency silly but confusing question

[meesumkazmi]

Hi Everyone,

I came across this question in a Kaplan test

If ab ≠ 0 and a3b = ab3, then which of the following must be true?

I a = b
II a = -b
III ab = 1

A. None
B. I only
C. III only
D. I and III only
E. I, II, and III

It seems I am making some silly mistake here

[meesumkazmi]

The quoted answer is "A. None"

[Anurag@Gurome]

If ab ≠ 0 and a3b = ab3, then which of the following must be true?

I a = b
II a = -b
III ab = 1

I assume a3b and ab3 are actually a³b and ab³.

--> a³b = ab³
--> (a³b - ab³) = 0
--> ab(a² - b²) = 0
--> ab(a - b)(a + b) = 0

As ab ≠ 0, either (a - b) or (a + b) or both must be equal to zero and ab can have any value.

If, (a - b) = 0, then a = b
If, (a + b) = 0, then a = -b
And if, both of them are equal to zero, then a = b = 0, which is again not possible as ab ≠ 0

Hence, either a = b or a = -b
Hence, either I or II is true.
But none of them must be true.

The correct answer is A.

[ronnie1985]

It has to be (A) None

As we are not sure if a = b or a = -b