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If a^5*b^6*c^7>0, is a*b>0?

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Source: — Data Sufficiency |

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by Jay@ManhattanReview » Mon Oct 30, 2017 2:54 am
LUANDATO wrote:$$If\ \ a^5b^6c^7>0,\ is\ ab>0?$$

$$1.b<0$$

$$2.c>0$$

The OA is C.

Can any expert assist me with this DS question, please? I don't have it clear. Thanks.
Since a^5b^6c^7 > 0, we have b^6 > 0, and b's exponent is 6, an even number, b can be positive or negative. Moreover, since the product of a^5 (odd exponent) and c^7 (odd exponent) is greater than 0, either a and c both are positive or both are negative,

(1) b < 0

Case 1: If a is positive ab < 0, the answer is No.
Case 2: If a is negative ab > 0, the answer is Yes. No unique answer. Insufficient.

(2) c > 0

As discussed above that the signs of a and c are same, since c > 0, a > 0.

Case 1: If b is positive ab > 0, the answer is Yes.
Case 2: If b is negative ab < 0, the answer is No. No unique answer. Insufficient.

(1) and (2) combined:

From (1), b < 0 and from (2), a > 0, thus ab < 0. the answer is No. A unique answer. Sufficient.

The correct answer: C

Hope this helps!

-Jay

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by GMATGuruNY » Mon Oct 30, 2017 3:46 am
LUANDATO wrote:$$If\ \ a^5b^6c^7>0,\ is\ ab>0?$$

$$1.b<0$$

$$2.c>0$$
a�b�c� > 0 implies that a, b and c are all NONZERO.
A nonzero value raised to an even power is positive.
Thus, we safely divide the inequality above by any even power of a, b and c.
If we divide each side by a�b�c�, we get:
a�b�c�/a�b�c� > 0/a�b�c�
ac > 0.

The resulting inequality indicate that a and c have the SAME SIGN.
The inequality in the question stem -- ab > 0 -- is valid only if a and b have the same sign.
Thus, the answer to the question stem will be YES only if a, b and c all have the same sign.
Question stem, rephrased:
If a and c have the same sign, do a, b and c all have the same sign?

Statement 1:
No information about the sign of a and c.
INSUFFICIENT.

Statement 2:
Since c is negative -- and a and c have the sign sign -- a is negative.
No information about the sign of b.
INSUFFICIENT.

Statements combined:
Statement 1 indicates that b is positive.
Statement 2 indicates that a and c are negative.
Thus:
a, b and c do NOT all have the same sign, with the result that the answer to the question stem is NO.
SUFFICIENT.

The correct answer is C.
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