If abc = b^3 , which of the following must be true?

[BTGModeratorVI]

If abc = b^3 , which of the following must be true?

I. ac = b^2
II. b = 0
III. ac = 1

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

Answer: A
Source: Kaplan

[Brent@GMATPrepNow]

If abc = b^3 , which of the following must be true?

I. ac = b^2
II. b = 0
III. ac = 1

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

Answer: A
Source: Kaplan

Let's answer this question by Process Of Elimination.

I. ac = b²
Must this be true? No.
Consider the case where a = 1, b = 0 and c = 1
This set of values satisfies the given equation (abc = b³),
HOWEVER it is not the case that ac = b²
Plug in the values to get (1)(1) = 0² - NOT TRUE

II. b = 0
Must this be true? No.
Consider the case where a = 1, b = 1 and c = 1
This set of values satisfies the given equation (abc = b³),
HOWEVER it is not the case that b = 0

NOTE: At this point, we need not examine statement III, because none of the answer choices state that only statement III must be true.

However, let's examine statement III for "fun"

III. ac = 1
Must this be true? No.
Consider the case where a = 0, b = 0 and c = 0
This set of values satisfies the given equation (abc = b³),
HOWEVER it is not the case that ac = 1

Answer: A

[Scott@TargetTestPrep]

If abc = b^3 , which of the following must be true?

I. ac = b^2
II. b = 0
III. ac = 1

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

Answer: A

Solution:

abc = b^3

abc - b^3 = 0

b(ac - b^2) = 0

(b = 0) or (ac - b^2 = 0 → ac = b^2)

Since it’s either b = 0 or ac = b^2, so neither one is a must true statement. Also, nowhere we can conclude that ac = 1. So none of the statements is a must true.

Answer: A