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# For nonnegative integers a, b, and c, what is the value of the product

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For nonnegative integers a, b, and c, what is the value of the product  [#permalink]

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Updated on: 02 Mar 2015, 03:04
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70% (01:10) correct 30% (01:16) wrong based on 233 sessions

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For nonnegative integers a, b, and c, what is the value of the product abc?

(1) ab=bc
(2) a≠c

I cannot understand the explanation given by veritas prep. Need help on this one. will post the explanation

Originally posted by sayansarkar on 02 Mar 2015, 02:40.
Last edited by Bunuel on 02 Mar 2015, 03:04, edited 1 time in total.
Renamed the topic, edited the question, added the OA and moved to DS forum.
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Re: For nonnegative integers a, b, and c, what is the value of the product  [#permalink]

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02 Mar 2015, 03:10
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For nonnegative integers a, b, and c, what is the value of the product abc?

(1) ab=bc;
ab - bc = 0;
b(a - c) = 0;
Either b =0 or a = c.
If b = 0, then abc = 0 but if b ≠ 0, then the value of abc will depend on the individual values of the variables. Not sufficient.

(2) a≠c. Not sufficient.

(1)+(2) Since from (2) a≠c, then from (1) we'd have that b = 0, making abc equal to 0, irrespective of the values of a and c. Sufficient.

Hope it's clear.
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Re: For nonnegative integers a, b, and c, what is the value of the product  [#permalink]

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02 Mar 2015, 03:30
Thank you Bunuel. Now its clear. I missed the point there
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Re: For nonnegative integers a, b, and c, what is the value of the product  [#permalink]

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05 Mar 2015, 12:44
Bunuel- you stated that "Either b =0 or a = c." when we get to the point of b(a-c)=0 is that always the case, where only one is true, when we have an equation that looks like that? So for example (just making this up)...

12A=9AB
12A-9AB=0
factor out the 3A...
3A(4-3B)=0
So then..only one of these solutions can be true?
A=0 OR B=4/3
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Re: For nonnegative integers a, b, and c, what is the value of the product  [#permalink]

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05 Mar 2015, 12:55
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Hi soniasawhney,

Both possibilities COULD be true, but it's not grammatically correct (nor mathematically correct) to state:

B=0 AND A=C (as both DO NOT need to be true for the product of B(A-C) to equal 0 - only one of them needs to be true).

If you were taking the notes, then you might find it useful to write....

B=0 or A=C OR BOTH

12A = 9AB
12A - 9AB = 0
3A(4 - 3B) = 0

A=0 or B=4/3 OR BOTH

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Re: For nonnegative integers a, b, and c, what is the value of the product  [#permalink]

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05 Mar 2015, 14:28
ah that makes sense- thank you!
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Re: For nonnegative integers a, b, and c, what is the value of the product  [#permalink]

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09 Jul 2017, 06:44
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sayansarkar wrote:
For nonnegative integers a, b, and c, what is the value of the product abc?

(1) ab=bc
(2) a≠c

I cannot understand the explanation given by veritas prep. Need help on this one. will post the explanation

(1) ab=bc

$$ab = bc$$

$$ab - bc = 0$$

$$b(a - c) = 0$$

Either $$b = 0$$ or $$a = c$$

We will get multiple answers based on values of b and a or c.

If b = 0 then we will the value of equation as ZERO, however if the a = c then we will get value, based on a or c.

Hence, (1) ===== is NOT SUFFICIENT

(2) a≠c

This does not give us any values of a or b or c

Hence, (2) ===== is NOT SUFFICIENT

Combining (1) & (2)

We get that a≠c which means $$b = 0$$ as per (1) and this gives us the value of ZERO foe abc.

Hence, (1) & (2) ===== is SUFFICIENT

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Re: For nonnegative integers a, b, and c, what is the value of the product  [#permalink]

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10 Jul 2017, 10:59
sayansarkar wrote:
For nonnegative integers a, b, and c, what is the value of the product abc?

(1) ab=bc
(2) a≠c

I cannot understand the explanation given by veritas prep. Need help on this one. will post the explanation

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

There are 3 variables and 0 equation. Thus the answer E is most likely.

Condition 1)
$$ab = bc$$ is equivalent to $$ab - bc = 0$$ or $$b(a-c) = 0$$.
Thus $$b = 0$$ or $$a = c$$.
If $$b = 0$$, then we have $$abc = 0$$.
However, if $$b \ne 0$$ and $$a = c$$, we have the case $$a = b = c = 1$$ with $$abc = 1$$.
Since the answer is not unique, this is not sufficient.

Condition 2)
If $$a = 1$$, $$b = 1$$, $$c = 2$$, then $$abc = 2$$.
If $$a = 1$$, $$b = 2$$, $$c = 3$$, then $$abc = 6$$.

Since the answer is not unique, this is not sufficient either.

Conditions 1) & 2)
From the condition 1), $$ab = bc$$ is equivalent to $$ab - bc = 0$$ or $$b(a-c) = 0$$.
Thus $$b = 0$$ or $$a = c$$.
However, $$b = 0$$, since $$a \ne c$$ by the condition 2)
Thus $$abc = 0$$.

For cases where we need 3 more equations, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80 % chance that E is the answer, while C has 15% chance and A, B or D has 5% chance. Since E is most likely to be the answer using 1) and 2) together according to DS definition. Obviously there may be cases where the answer is A, B, C or D.
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Re: For nonnegative integers a, b, and c, what is the value of the product  [#permalink]

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05 Sep 2019, 23:13
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Re: For nonnegative integers a, b, and c, what is the value of the product   [#permalink] 05 Sep 2019, 23:13
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