For integers a, b, and c, if ab = bc, then which of the foll

Question

For integers a, b, and c, if ab = bc, then which of the following must also be true?

A. a = c
B. a^2*b=b*c^2
C. a/c = 1
D. abc > bc
E. a + b + c = 0

Bunuel · Accepted Answer

ab = bc  -->  ab-bc=0  -->  b(a-c)=0 -->  b=0  or  a=c .

A. a = c. If  b=0 , then this option is not  necessarily  true.

B. a^2*b=b*c^2 -->  b(a^2-c^2)=0  -->  b(a-c)(a+c)=0 . Now, since  b=0  or  a=c , then  b(a-c)(a+c)  does equal to zero. So, we have that this options must be true.

C. a/c = 1. If  b=0 , then this option is not  necessarily  true.

D. abc > bc. If  b=0 , then this option is not true.

E. a + b + c = 0. If  b=0 , then this option is not  necessarily  true (if b=0 then a+c can take any value this option is not necessarily true.).

Answer: B.

mattce · Answer

The point here is that analyzing the initial equation ab = bc, we know that either b = 0 or a = c.
Plugging in either of these values results in b(a-c)(a+c) being equal to zero.
I think you're confusion comes from extracting "a = -c" from "b(a-c)(a+c) = 0" as opposed to extracting information from the original equation ab = bc (which is what we need to do).

Continuing from above:
So recall that from the first equation, b = 0 or a = c.
The expanded term b(a-c)(a+c) is an alternate expression for the second option; i.e, the fact that both of the possible necessary truths (b = 0; a = c) lead to b(a-c)(a+c) = 0 means that the equivalent expression, a^2 * b = b * c^2 must also be true.

I hope that's clear!

piealpha · Answer

b(a-c)=0 => b=0 or a=c

You can see B in an easier way:
Inequalities hold when we raise them to power on both sides. so a^2= c^2. => b.a^2= b.c^2. Thus B follows right away.

I agree with other explainations by Bunuel.

dhlee922 · Answer

for answer choice B, could you also say that if a = -c it also equals zero?

Also, why do all of the possible answers equaling zero in choice B mean that B must be true?

dhlee922 · Answer

i'm still a little confused by the answer choices.

is it because the question stem equates to B=0  OR  A=C

so for answer choice A, A doesnt have to equal C because the B equaling 0 will just make the whole expression 0

but then for answer choice B, since B=0  OR  A=C, since 1 or the 3 groups in parentheses will be 0, it will make the whole expression 0, therefore B must be true?

Zarrolou · Answer

Yes, exactly. We know that or b=0 or a-c=0 ( or both) .

We can rewrite B as  b(a-c)(a+c) , and since at least one of the terms is 0, the whole expression is 0.

Bunuel · Answer

Yes, for A, if b=0, then a may or may not equal to c.

As for B, since b=0 or a=c (a-c=0), then b(a-c)(a+c)=0 must be true, since either the first or the second multiple (or both) is 0.

Hope it's clear.

dhlee922 · Answer

yes, thank you everyone for your replies! geez, i find this one to be pretty tricky, but it's only 600-700 level, ugh

echo900 · Answer

Hi everyone,

This question is messing with everything I know (which is good). Isn't it legal to divide both parts of the equation by the same value? In this case, by b:

ab = bc --> (ab)/b = (bc)/b --> a = c

What am I doing wrong here?

Thanks!

dcummins · Answer

Official Explanation: 
 Correct Answer: B

Choices A and C are essentially identical, and should be regarded with suspicion. Upon further review, if b were to equal 0, then a need not necessarily equal b. Try numbers: a = 1, b = 0, c = 2, then (1)(0) = (0)(2), but 1 does not equal 2. Because b could equal zero, answer choice D is also incorrect, as the two sides of the inequality would both equal zero, and therefore be equal. Answer choice E is also not necessarily true, as demonstrated by the trial numbers above. Answer choice B is correct, as it is true both if a = c, or if b is equal to zero.

eantolovic · Answer

Clarity did not come to me from this thread until I read your explanation, and I'm very grateful that you took the time to post it.

julianaescobar · Answer

Hi Bunuel- how come we can't start out by dividing both sides by b? I dont think to set everything equal to 0. Thanks

julianaescobar · Answer

I have the same question

JiddiHuMain · Answer

@permalink,

Actually this is very good question, it checks your reasoning skills.
Here is solution -

Given : ab = bc , a,b,c are integers,-
let's solve step by step - 
ab = bc ---> b(a-c) = 0 ------> either b = 0 or a - c = 0,

hence ,
option (A) a = c must be true ???? lets' check a-c = 0 how we are sure from given information --- because here is no idea about what is value of b ... may be from given information b = 0 and a = 4 , c = 5 , then how a = c must be true ??? hence..here is no clarity...no this option may be true but not must be true. so option(A) is not correct.

before option(B) will check rest options -

Option(C) a/c = 1 must be ?????? ..how we can are sure.. we don't have any idea about value of b..b may be 0 , and maybe a= 4, c= 7....so a/c =1 ..not possible..hence..here are this option is not correct.

Option(D) abc>bc...????? ...if b= 0..then ???...this option is also not correct.

Option(E) --- out of scope.

let's come for option(B)-

here ba^2 =bc^2 -----> b(a-c)(a+c) = 0 ...and form given information either a=c or b=0..hence...if b= 0 or a -c = 0 then satisfy --lhs =rhs...hence this option must be true.

hence option B is correct.

I hope you understand.

If like my solution please give kudos...

Basshead · Answer

If  ab = bc , then

ab - bc = 0

b(a-c) = 0

b = 0  or  a = c

Lets look at the statements:

A. Doesn't have to be true. b could equal 0.

B.  a^2*b=b*c^2

a^2*b - c^2*b = 0

b(a^2 - c^2) = 0

b(a+c)(a-c) = 0
 
Must be true.

C. Doesn't have to be true. b could equal 0.

D.  abc > bc

abc - bc >0

bc(a-1) > 0

Not necessarily true.

E. Cleary wrong.

Answer is B.

ocelot22 · Answer

ab=bc factors to b(a-c)=0. This means b = 0 or a-c =0 ---> a=c, or both arguments are true.

A. a=c not neccessarilly true if b=0 elim
B. a^2b=bc^2---> a^2b-bc^2=0--> b(a^2-c^2) = 0---> b(a+c)(a-c) = 0---> (a+c)*b*(a-c)=0 since we get a scale multiple of our original argument, this must also be true.

OA is B

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.

For integers a, b, and c, if ab = bc, then which of the foll

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

For integers a, b, and c, if ab = bc, then which of the foll

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