Is The Product Of a And b Equal To 1? GMAT Data Sufficiency

Question: Is the product of a and b equal to 1?

(1) a∗b∗a = a
(2) b∗a∗b = b

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient

“Is the product of a and b equal to 1?” is a topic of the GMAT Quantitative reasoning section of GMAT. The GMAT Data Sufficiency question is characterized by a problem statement and two factual statements. This specific GMAT data sufficiency question considers the skill of the candidates in calculating the quantitative problems. The smart wording used in these questions is the most difficult portion that candidates usually overlook. GMAT Quant section constitutes 31 multiple-choice questions. GMAT data sufficiency contains 15 questions out of these 31 GMAT quant questions. The candidates can brush up their quantitative knowledge by answering more questions from the book named “GMAT Data Sufficiency Prep Course”.

Solution and Explanation:

Approach Solution 1:

The problem statement asks to find out if the product of a and b is equal to 1.

1. The statement states that a∗b∗a = a
   That is, we can say, a^2* b = a
   Therefore, in the next step, we can write, a^2* b−a = 0;
   Therefore, it can be rewritten as a(ab - 1)= 0, where either a= 0 (b=any value, including zero), so in this case ab = 0 ≠1 Or, ab = 1.
   Thus we are getting two distinct answers. Hence the statement is not sufficient.
2. The statement states that b∗a∗b = b
   That is, we can say, a*b^2 = a
   Therefore, in the next step, we can write, a*b^2−b= 0
   Therefore, it can be rewritten as b(ab−1)=0 where either b=0 (a=any value, including zero), so in this case ab = 0 ≠1 Or ab=1

Thus we are getting two distinct answers. Hence the statement is not sufficient.

Combining both statements, we get,
Either a=b= 0, therefore, in this case, ab=0≠1
Therefore, the answer to this problem is No.
Or, ab=1, therefore, the answer to this problem is yes.
Thus we are getting two distinct answers. Hence both statements together are not sufficient.

Correct Answer: (E)

Approach Solution 2:

The problem statement asks to find out if the product of a and b is equal to 1.

1. The statement states the equation: a∗b∗a = a
   If the value of a can be 0 or 1, then the equation of the statement still holds;
   Hence the statement alone is not sufficient.
2. The statement expresses the equation: b∗a∗b = b
   If the value of b can be 0 or 1, then the equation of the statement still holds;
   Hence the statement alone is not sufficient.
   Combining both equations of the statements, we get,
   If the value of a = b = 1, then the answer to this problem is Yes.
   If the value of a = b = 0, then the answer to this problem is No.
   Hence, both statements together are not sufficient.

Correct Answer: (E)

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