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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

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

Re: For integers a, b, and c, if ab = bc, then which of the foll [#permalink]

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11 Jun 2013, 20:47

Bunuel wrote:

emmak wrote:

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

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

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.

Answer: B.

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?

Re: For integers a, b, and c, if ab = bc, then which of the foll [#permalink]

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11 Jun 2013, 21:21

1

This post received KUDOS

dhlee922 wrote:

Bunuel wrote:

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

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.

Answer: B.

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

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).

dhlee922 wrote:

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

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.

Re: For integers a, b, and c, if ab = bc, then which of the foll [#permalink]

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12 Jun 2013, 15:29

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?

Re: For integers a, b, and c, if ab = bc, then which of the foll [#permalink]

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12 Jun 2013, 15:34

dhlee922 wrote:

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?

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.
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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?

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.

Re: For integers a, b, and c, if ab = bc, then which of the foll [#permalink]

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30 Jul 2014, 06:53

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Re: For integers a, b, and c, if ab = bc, then which of the foll [#permalink]

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19 Jun 2016, 01:47

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