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

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Re: If abc = b^3 , which of the following must be true? [#permalink]
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nikhilsrl wrote:
If abc = b³, which of the following must be true?

I. ac = b²
II. b = 0
III. ac = 1

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

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

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Re: If abc = b3 , which of the following must be true? [#permalink]
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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
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Re: If abc = b3 , which of the following must be true? [#permalink]
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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
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Re: If abc = b^3 , which of the following must be true? [#permalink]
Thanks Bunuel.. you are awesum
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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
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Re: If abc = b^3 , which of the following must be true? [#permalink]
Why is it wrong to divide both sides by b at the start?
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Re: If abc = b^3 , which of the following must be true? [#permalink]
Why is it wrong to divide both sides by b at the start?
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Re: If abc = b^3 , which of the following must be true? [#permalink]
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Why is it wrong to divide both sides by b at the start?

Because b can be 0 and we cannot divide by 0.
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Re: If abc = b^3 , which of the following must be true? [#permalink]
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Why is it wrong to divide both sides by b at the start?

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.
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Re: If abc = b^3 , which of the following must be true? [#permalink]
nikhilsrl wrote:
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

This is from Kaplan CAT. I am not quite sure how they arrived at the 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
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Re: If abc = b^3 , which of the following must be true? [#permalink]
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Why is it wrong to divide both sides by b at the start?

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
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Re: If abc = b^3 , which of the following must be true? [#permalink]
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.
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Re: If abc = b^3 , which of the following must be true? [#permalink]
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nikhilsrl wrote:
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

This is from Kaplan CAT. I am not quite sure how they arrived at the 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
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Re: If abc = b^3 , which of the following must be true? [#permalink]
nikhilsrl wrote:
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

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.

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Re: If abc = b^3 , which of the following must be true? [#permalink]
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nikhilsrl wrote:
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

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

Quote:
Hi can you pls tell me if b=0 then why is it no ?

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.
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Re: If abc = b^3 , which of the following must be true? [#permalink]
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