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# M01-36: Is the product of a and b equal to 1?

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Manager
Joined: 12 May 2012
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Concentration: General Management, Operations
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Kudos [?]: 57 [0], given: 14

M01-36: Is the product of a and b equal to 1? [#permalink]  09 Jun 2012, 07:49
Is the product of a and b equal to 1?
(1) a∗b∗a=a
(2) b∗a∗b=b

OA:
[Reveal] Spoiler:
Question: is ab=1?

(1) a^2*b=a → a^2*b−a=0 → a(ab−1)=0 → either a=0 (and b=any value, including zero) so in this case ab=0≠1 OR ab=1. Two different answers, not sufficient.

(2) a*b^2=b → a*b^2−b=0 → b(ab−1)=0 → either b=0 (and a=any value, including zero) so in this case ab=0≠1 OR ab=1. Two different answers, not sufficient.

(1)+(2) either a=b=0, so in this case ab=0≠1 and the answer to the question is NO, OR ab=1 and the answer to the question is YES. Two different answers, not sufficient.

I had figured that Each alone is insufficient.

But if we have two statements with only one common solution, then shouldnt that be the answer?
from 1 we get: either a=0 or ab=1
from 2 we get: either b=0 or ab=1
combining both we get only one common option, ab=1 - Sufficient
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Re: M01-36: Is the product of a and b equal to 1? [#permalink]  09 Jun 2012, 07:56
Expert's post
manulath wrote:
Is the product of a and b equal to 1?
(1) a∗b∗a=a
(2) b∗a∗b=b

OA:
[Reveal] Spoiler:
Question: is ab=1?

(1) a^2*b=a → a^2*b−a=0 → a(ab−1)=0 → either a=0 (and b=any value, including zero) so in this case ab=0≠1 OR ab=1. Two different answers, not sufficient.

(2) a*b^2=b → a*b^2−b=0 → b(ab−1)=0 → either b=0 (and a=any value, including zero) so in this case ab=0≠1 OR ab=1. Two different answers, not sufficient.

(1)+(2) either a=b=0, so in this case ab=0≠1 and the answer to the question is NO, OR ab=1 and the answer to the question is YES. Two different answers, not sufficient.

I had figured that Each alone is insufficient.

But if we have two statements with only one common solution, then shouldnt that be the answer?
from 1 we get: either a=0 or ab=1
from 2 we get: either b=0 or ab=1
combining both we get only one common option, ab=1 - Sufficient

Solution gives another possible case which satisfies both statements and gives a No answer to the question whether ab=1, namely a=b=0.
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Re: M01-36: Is the product of a and b equal to 1? [#permalink]  11 Jun 2012, 22:05
Is the product of a and b equal to 1?
(1) a∗b∗a=a
(2) b∗a∗b=b

1) not too sure, a could be 0
2) not too sure, b could be 0
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Re: M01-36: Is the product of a and b equal to 1? [#permalink]  14 Jun 2012, 02:44
Hi Bunuel, we cant divide by 'a' in the first condition or for that fact 'b' in the second equation, as we do not know the sign of the a and b. Am i right?
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Re: M01-36: Is the product of a and b equal to 1? [#permalink]  14 Jun 2012, 02:57
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Expert's post
pavanpuneet wrote:
Hi Bunuel, we cant divide by 'a' in the first condition or for that fact 'b' in the second equation, as we do not know the sign of the a and b. Am i right?

Not quite. We cannot divide by a because a can be zero and division by zero is not allowed.

Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We can not divide by zero. So, if you divide (reduce) a∗b∗a=a by a you assume, with no ground for it, that a does not equal to zero thus exclude a possible solution.

As for the sign: it becomes important with inequalities. For example if we have xy>x we cannot divide (reduce) by x since we don't know it's sign: if x>0 then we would have y>1 BUT if x<0 then we would have y<1 (flip the sign of the inequality when dividing/multiplying by negative value).

So, never multiply or divide inequality by a variable (or by an expression with variable) unless you are sure of its sign since you do not know whether you must flip the sign of the inequality.

Hope it's clear.
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Re: M01-36: Is the product of a and b equal to 1? [#permalink]  16 Jun 2012, 08:44
Rephrasing of the two statements yield
1) a=0 or ab=1 (not sufficient since ab=1 or ab=0)
2) b=0 or ab=1 (not sufficient since ab=1 or ab=0)

Combining yields same result, videlicit: ab=1 or ab=0

Thus E.

Cheers.
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Re: M01-36: Is the product of a and b equal to 1?   [#permalink] 16 Jun 2012, 08:44
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# M01-36: Is the product of a and b equal to 1?

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