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Integer question

by AkiB » Thu Jul 03, 2014 4:30 am
For integers a, b, and c, if ab = bc, then which of the following must also be true?

A) a = c
B) a^2b=bc^2
C) ac = 1
D) abc > bc
E) a + b + c = 0

OA-B
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by GMATGuruNY » Thu Jul 03, 2014 4:54 am
AkiB wrote:For integers a, b, and c, if ab = bc, then which of the following must also be true?

A) a = c
B) a^2b=bc^2
C) ac = 1
D) abc > bc
E) a + b + c = 0
ab = bc
ab - bc = 0
b(a-c) = 0.

Implication:
Either b=0 or a-c=0.
Given this constraint, prove that four of the answer choices DON'T have to be true.

Case 1: b=0, a=2, c=1
In this case, A, C, D and E are not true.
Eliminate A, C, D and E.

The correct answer is B.
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by Brent@GMATPrepNow » Thu Jul 03, 2014 5:35 am

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

A) a = c
B) a²b = bc²
C) ac = 1
D) abc > bc
E) a + b + c = 0
Mitch's approach (where he eliminates A, C, D and E to show that B must be correct) is perfect.
This next approach shows exactly why B is correct.

Take ab = bc
Rearrange to get: ab - bc = 0
Factor to get: b(a - c) = 0
From this, we can conclude that b = 0 OR a - c = 0.
In other words b = 0 OR a = c

Now check answer choice B.
Must it be true that a²b = bc²?
Yes!
Rearrange to get a²b - bc² = 0
Factor: b(a² - c²) = 0
Factor more: b(a + c)(a - c) = 0

Now we already know that b = 0 OR a = c
If b = 0, then b(a + c)(a - c) must equal 0
If a = c, then b(a + c)(a - c) must equal 0

Since BOTH possible cases (b = 0 or a = c) result in answer choice B being true, the correct answer is B

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by GMATinsight » Thu Jul 03, 2014 6:26 am
For integers a, b, and c, if ab = bc, then which of the following must also be true?

A) a = c
B) a^2b=bc^2
C) ac = 1
D) abc > bc
E) a + b + c = 0
It's given that ab = bc

The mistake that many students make is cancelling the variable "b" which is available on on both sides so PLEASE DON'T CANCEL VARIABLES UNTIL THEIR SIGN IS KNOWN AS POSITIVE.

ab-bc = 0
b (a-c) = 0

Product of two is zero when one of the two numbers is zero

Therefore, b=0 or a=c

Therefore
Option A is incorrect as no certainty is there if b=0
Option B is correct as a²b = bc² is same as b(a²- c²) which means either b=0 or lal=lcl which is always true based on the inference from the questions. therefore CORRECT ANSWER CHOICE
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by ceilidh.erickson » Fri Jul 04, 2014 9:37 am
Rather than solving algebraically, as other experts have done, you could think about this conceptually: most students will interpret ab = bc to mean that a = c, and will choose answer A. For one thing, this should feel to easy to be a GMAT question! For another, you should always test all of the answer choices on a "which MUST be true" question. You would also have gotten a "yes" answer for B, at which point you would have to consider - is there any case in which ab = bc, but a does not equal c? Remember to always consider 0! That's often the "dealbreaker."

Other examples of 0 as a dealbreaker:

xy = x^2 --> x=y or x=0
|n| = -n --> n < 0 or n=0
2x/3 = integer --> x is divisible by 3 or x=0 (technically 0 is also divisible by 3, but you'd still want to consider it separately)
etc
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by Matt@VeritasPrep » Sun Jul 06, 2014 6:42 pm
GMATinsight wrote:
The mistake that many students make is cancelling the variable "b" which is available on on both sides so PLEASE DON'T CANCEL VARIABLES UNTIL THEIR SIGN IS KNOWN AS POSITIVE.
This is totally wrong. You only need to know that the variable is nonzero - dividing both sides of an equation by a real, nonzero variable won't muck anything up.

In an inequality you need to consider the sign, but even then you aren't constrained to positive numbers.
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by GMATinsight » Sun Jul 06, 2014 7:10 pm
This is totally wrong. You only need to know that the variable is nonzero - dividing both sides of an equation by a real, nonzero variable won't muck anything up.
I hope you can recognize that you can misguide the readers by posting such a comment under the tag of Instructor.

I have mentioned a comment that generalized Equation and Inequality both because I don't teach by rules, I teach by least rules but more logics. Whereas It seems your style to teach every case separately and for that if you tag something "Correct" as "wrong" then it definitely UPSETS me.

Mind your language. Think before posting comments. And keep your unwanted suggestion with you please. It's a request.
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by Matt@VeritasPrep » Sun Jul 06, 2014 8:09 pm
GMATinsight wrote:I hope you can recognize that you can misguide the readers by posting such a comment under the tag of Instructor.

I have mentioned a comment that generalized Equation and Inequality both because I don't teach by rules, I teach by least rules but more logics. Whereas It seems your style to teach every case separately and for that if you tag something "Correct" as "wrong" then it definitely UPSETS me.

Mind your language. Think before posting comments. And keep your unwanted suggestion with you please. It's a request.
I didn't mean to upset you, but your statement was incorrect enough to warrant a response: students often take highlighted statements to heart and are apt to remember pronouncements like "You can't cancel a variable unless you know its sign is positive". I'm not splitting hairs here: what you said just ISN'T TRUE without significant qualification, and any student who adopted that statement as a rule could then be prone to major mistakes on simple questions, be they on the GMAT or in a MBA quant class. I can think of dozens of DS questions that a student could get wrong were he to take that dictum literally, so I felt it was important to address it.

Not sure what you mean by 'mind your language': there wasn't anything profane in my remarks.
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by GMATinsight » Sun Jul 06, 2014 11:51 pm
Matt@VeritasPrep Wrote:
I'm not splitting hairs here: what you said just ISN'T TRUE without significant qualification, and any student who adopted that statement as a rule could then be prone to major mistakes on simple questions, be they on the GMAT or in a MBA quant class. I can think of dozens of DS questions that a student could get wrong were he to take that dictum literally, so I felt it was important to address it.
If you are not spitting hair and you have dozens of examples that can go wrong if my highlighted instruction is followed (that means without canceling variables, it leads to wrong answer and it leads to right answer if variables are cancelled), Then please mention one such example here.
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by Brent@GMATPrepNow » Mon Jul 07, 2014 6:33 am
Hi Bhoopendra,

I've reread this thread a few times, and I have a feeling that you may be inadvertently defending something that isn't true.

In your first post, you wrote:
The mistake that many students make is cancelling the variable "b" which is available on on both sides so PLEASE DON'T CANCEL VARIABLES UNTIL THEIR SIGN IS KNOWN AS POSITIVE.

Just to be clear, are you suggesting that we cannot divide both sides of an equation (or an inequality for that matter) by a variable unless we are certain that the variable is positive? If so, there is no end to the problems that test-takers would experience if they followed that advice.

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by GMATinsight » Mon Jul 07, 2014 6:40 am
Hi Brent,

No, I don't mean that you can't divide the equation on both sides until it's positive.

I only meant that in order to be sure of having made no mistake, it's always safe for students not to divide the equation by variable as in some cases it doesn't cause any problem (Equations where it's given that the variable is non zero) but in certain other cases (inequations) it may affect the solution if the sign of the variable is unknown.

Also, Since GMAT tests on very critical aspects of problem and tries to pose information in such a manner that many students make silly errors of not noticing whether x is positiove or x is Integer, or x is non zero therefore in order to safeguard them from any probable mistake, not canceling the variable is an option that can be exercised because to come to the conclusion it becomes necessary to read the question carefully again if any information has been missed by a student.
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by Matt@VeritasPrep » Mon Jul 07, 2014 9:09 am
GMATinsight wrote:If you are not spitting hair and you have dozens of examples that can go wrong if my highlighted instruction is followed (that means without canceling variables, it leads to wrong answer and it leads to right answer if variables are cancelled), Then please mention one such example here.
Following your advice on literally ANY QUESTION that involves dividing both sides of an inequality by a variable COULD lead a student to the wrong answer. This is especially true in Data Sufficiency, where a student following your statement would assume that knowing x > 0 is a NECESSARY CONDITION for dividing by x.
I only meant that in order to be sure of having made no mistake, it's always safe for students not to divide the equation by variable as in some cases it doesn't cause any problem (Equations where it's given that the variable is non zero) but in certain other cases (inequations) it may affect the solution if the sign of the variable is unknown.
I wouldn't give this advice either: it deprives students of a useful, coherent approach to equations and inequalities.

At the GMAT level (where you don't consider many equations beyond quadratics and linear equations), I'd go with a much simpler set of practices.

EQUALITIES::
If you want to divide both sides of an equation by x, first check to see if x = 0 is a solution to that equation. Then divide both sides by x to find any other solutions to the equation.

For instance, consider the equation x² = 9x. We'll start by checking x = 0, which is a solution, since 0² = 9*0. Then, to find any other solutions, we'll divide both sides by x, obtaining x = 9. Hence the equation has two solutions: x = 0 and x = 9.

Another example would be the equation rs = st. We'll start by checking s = 0, which is a solution, since r*0 = 0*t. Now we'll divide both sides by s to find the other solution: r = t.

INEQUALITIES::
Consider the consequences of dividing by a positive value of x AND a negative value of x. Also remember that if x is zero, you can't divide by x, so check x = 0 if appropriate.

For instance, say we have the inequality 3x > y.

If x > 0, we have 3 > y/x.

If x < 0, we have 3 < y/x.

If x = 0, then we have 0 > y.

Now we know that one of these conclusions is true, so we can answer immediately once we know the sign of x ... but we might be able to answer anyway, depending on what the problem is asking.
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by GMATinsight » Mon Jul 07, 2014 10:49 am
Hi Matt,
Considering your substantial experience as GMAT trainer against only 10 years of my experience as GMAT trainer, I respect your opinion.

Since this discussion is not adding any value to us or any other aspirant following this post, I choose to take myself out of this discussion.

Still I haven't got any CONCRETE example that would have enlightened me on the subject of this discussion and proven me wrong.

You would not suggest my method doesn't make my method INCORRECT. In-fact I never expected you to follow it, It was for GMAT aspirants.

All rules you have mentioned are correct. But I reiterate "NOT CANCELING the variables while solving Equation and Inequation never leads to the Incorrect answer". If it does, then I am waiting to be enlightened.
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by Matt@VeritasPrep » Mon Jul 07, 2014 4:15 pm
You would not suggest my method doesn't make my method INCORRECT
Well, let's explore. (I'll call it your Law rather than your Method, for clarity's sake: I can't integrate a method into the foundations of arithmetic, but I can define and explore the consequences of a law.)

Bhoopendra's Law, as I objected to it, was essentially defined as follows: "Both sides of an equation should be divided by a variable if and only if that variable is known to be positive". A corollary of Bhoopendra's Law is that one should not MULTIPLY both sides of an equation by a variable whose sign is unknown, as multiplication is "reciprocal division". (If I cannot divide both sides by 1/x, I also cannot multiply both sides by x, as these are identical operations for any nonzero x.) Here are a few problems that seem rather difficult to solve:

1:: x and y are integers such that y/x = 2. What is x in terms of y?

2:: If x/y = 3, is y = x/3?

3:: y is nonzero and z = xy. Is x = z/y?

This Law seems to have some pretty fascinating consequences - consider its effect on the transitivity of division, for instance! Mulling its consequences reminds me of Analysis 101, when we had to use the Peano Axioms to prove things like 1 ≠ 0.

{EDITED several times for clarity and diplomacy)
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