If ab 0 and a^3b = ab^3, then which of the following must be true?

Question

If ab ≠ 0 and  a^3b = ab^3 , 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

This is a  PS Butler Question

Check the links to other Butler Projects:

Data Sufficiency Butler  
  Critical Reasoning Butler  
  Reading Comprehension Butler  
  Data Insights Butler

ManifestDreamMBA · Answer

a^3b = ab^3 
 a^2 = b^2 
which means a = b or a = –b

I and II could be true but we don't know which one it is and both cannot be true at the same time

Additionally, ab = 1 is also not a must condition

There is no answer

GMATinsight · Answer

a^3b = ab^3 
i.e.  a^3b - ab^3 = 0 
i.e.  ab(a^2 - b^2) = 0

Since ab ≠ 0
therefore  (a^2 - b^2) = 0 
i.e.  (a - b)*(a + b) = 0

i.e. either a = b
or a = -b
but we can't be certain about any possible outcome hence

Answer: Option A

Related Video:

https://www.youtube.com/watch?v=kszOy_LCh2s

Abhiswarup · Answer

None should be answer

a3b=ab3
gives a2=b2 
a=+-b
ab unequals 0
nothing can be concluded precisely. Thus E should be answer.

Dereno · Answer

a3b - ab3 = 0
ab (a2 - b2) = 0
As per the above statement , ab =0 or (a2-b2) =0 
But, in the question it’s mentioned  ab is NOT equal to 0.

(a2-b2) must be zero. 
(a+b) (a-b) = 0 
Therefore, either  a = b or a=-b.

ab  can be any integer value. Can be 1 also.

If the question is could be true, then all three options can be true. Option E.

But,   MUST BE TRUE    Is the question asked. Hence  option A

Ayeka · Answer

a^3*b=a*b^3
(a^3*b)-(a*b^3)=0
ab(a^2-b^2)=0
ab ≠ 0 , so (a-b)(a+b)=0
either (a=b) or (a= -b) 
The question asks must be true, a can be either equal to b or -b, hence all the statements could be true but can’t be must be true 
I. a = b
II. a = –b
III. ab = 1

Answer: A

TC97 · Answer

The question ask for   must be true

a^3*b=a*b^3
ab(a^2)=ab(b^2)
a^2=b^2

(a,b) could be both positive, e.g. (10,10), and still fits the equation
(a,b) could be one positive and one negative, e.g. (10,-10); (-10,10), and still fits the equation
(a,b) could be both negative, e.g. (-10,-10), and still fits the equation

Therefore, (I) & (II) can be eliminated as the statement isn't "must be true" for the equation.

we can only conclude that ab=ab, but doesn't necessary means ab=1
Therefore, (III) can be eliminated as well.

Hence, option  (A) None  is the correct answer.

If ab 0 and a^3b = ab^3, then which of the following must be true?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If ab 0 and a^3b = ab^3, then which of the following must be true?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards