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

Question

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

This is from Kaplan CAT. I am not quite sure how they arrived at the answer.

Bunuel · Accepted Answer

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 abc = b^3 --> b(ac-b^2)=0 --> EITHER b=0 OR ac=b^2 , which means that NONE of the option MUST be true. For example if b=0 then ac can equal to any number (not necessarily to 0 or 1), so I and III are not always true, and if ac=b^2 then b can also equal to any number (not necessarily to 0), so II is not always true. Answer: A. More Must or Could be True Questions with solutions to practice: search.php?search_id=tag&tag_id=193 Hope it helps.

KarishmaB · Answer

Responding to a pm: I think I won't be able to take 'Must be true' for another week on my blog (Std Dev still not over). Hence, explaining this question right here. Given: abc = b^3 abc - b^3 = 0 b(ac - b^2) = 0 This tells us that either b = 0 OR ac = b^2 . At least one of these 2 'must be true'. So can I say b must be equal to 0? No! It is possible but it is also possible that instead, ac = b^2 . So can I say ac must be equal to b^2 ? No! It is possible but it is also possible that instead, b = 0. So can I say that b = 0 AND ac = b^2 ? No! It is possible but it is also possible that only one of them is true. So can I say that b = 0 OR ac = b^2 ? Yes, I can! One of them must be true. If one of the options were 'I or II', that would have been true. Updating the Blog Post link: https://anaprep.com/algebra-game-must-b ... questions/ https://anaprep.com/algebra-must-be-tru ... questions/

BrentGMATPrepNow · Answer

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

beyondgmatscore · Answer

abc = b^3 implies abc - b^3 = 0
or b*(ac-b^2)=0
which implies either b=0 or ac=b^2
so, either of them or both of them can be true, but none of them must be true. Answer A

imhimanshu · Answer

Hi Bunuel, 
Request you to post your reasoning. Here is how I approached this one-
abc = b^3
abc-b^3 =0
b(ac-b^2)=0
Hence either b=0 or ac=b^2

However b=0 is not a solution. Could you please shed some light on how to approach must be true or could be true questions.
Thanks
H

imhimanshu · Answer

Thanks Bunuel.. you are awesum :-)

Ashikurrahman · Answer

OA is definitely A. Folks may be little bit confused at option *i and iii* (3) tells us that ac=1. So what? Is b also 1 or equal to 0? Can't figure it out. Move further. If some part of multiplication is 0, there is no need to ans. That question. It doesnt satisfy the condition: abc=b^3.

NOTE:WHEN YOU ARE TACKLING "MUST BE TYPE" QUESTION, always try to find out " entire set" instead of "sub set" cz subset answers "what should be" rather than "what must be".
If you are an accountant, you may consider it equivalent to " Bad debt reserve".

Entire set= LCM
Suset =GCM.

Posted from my mobile device

STEMBusiness · Answer

Why is it wrong to divide both sides by b at the start?

STEMBusiness · Answer

Why is it wrong to divide both sides by b at the start?

Bunuel · Answer

Because b can be 0 and we cannot divide by 0.

KarishmaB · Answer

Another way to understand this:

When you divide by b, you are assuming that b is not 0. Is it a fair assumption? No. You are losing on a possible solution of the equation.

How can I make abc = b^3? I can do that by either making b = 0 or making ac = b^2.

But if you divide by b, you get only one solution: ac = b^2 and that means I must be true. Actually, the answer is none because there is another possibility and that is b = 0.

stonecold · Answer

HERE the question is asking us which statement must be true
Hence we can discard all of them as we can make each one of them untrue by taking the other leftover equations true.
Hence A

BrentGMATPrepNow · Answer

Here's an illustrative example: 
(5)(0) = (6)(0) TRUE
Now divide both sides by 0 to get: 5 = 6 NOT TRUE

Likewise, if xy = xz, we can't divide both sides by x and conclude that y = z, because it's possible that x = 0
So, while it's true that (0)(y) = (0)(z), we can't then conclude that y = z

narendran1990 · Answer

I considered three different integers:

When '1' is considered, Statement 1 & 3 are correct & when '3' is considered, statement 1 alone holds. Finally, considered a=-3,b=3 & c=3. None of the statements hold true.

Given that only 49% of the folks got this simple looking question right, understood that there is a trap.

NikGujju · Answer

abc=b^3
abc-b^3=0
b(ac-b^2)=0
either b=0 or ac=b^2

Definitely not 1 or 2 because EITHER

definitely not 3

JeffTargetTestPrep · Answer

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

KarishmaB · Answer

The question is looking for "MUST BE TRUE". Do we NEED b to be 0 for abc = b^3 to hold? No.

Say a = 1, b = 2 and c = 4

Then abc = 1*2*4 = 8
b^3 = 2^3 = 8

So even if b is not 0, abc can be equal to b^3.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

If abc = b^3 , 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 abc = b^3 , 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