What is the Value of xy? GMAT Data Sufficiency

Question: What is the value of xy?

1. \(3^x*5^y=75\)
2. \(3^{(x-1)(y-2)}=1\)

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.

Answer:
Solution with Explanation:

Approach Solution (1):

S1: \(3^x*5^y=75\)

If x and y are integers then as \(3^1*5^2=75\) then x = 1 and y = 2 but if they are not, then for any value of x, there will exist some non-integer y to satisfy given expression and vice-versa (example: if y = 1 then \(3^x*5^y=75\) hence \(3^x=15\) and x = some irrational number = 2.5)

Not sufficient

S2:\(3^{(x-1)(y-2)}=1\)

This statement basically says that (x – 1) (y – 2) = 0. So, either x = 1 and y is any number (including 2) or y = 2 and x is any number (including 1)
Not sufficient

(1) + (2) if from (2) x = 1 then from (1) \(3^x*5^y=3^1*5^y=75\) giving y = 2 and if from (2) y = 2 then from (1) \(3^x*5^y=3^1*25=75\) giving x = 1
Thus x = 1 and y = 2
Sufficient

Correct Option: C

Approach Solution (2):

S1: x = 1, y = 2

Or say x and y can be a fraction value as well
Hence insufficient

S2: \(3^{(x-1)(y-2)}=1\)

x and y not known
Hence insufficient
From (1) + (2), we get:
x = 1 and y = 2, we get:

\(3^{(x-1)(y-2)}=1\)

Hence sufficient

Correct Option: C

Approach Solution 3:
1. x=1,y=2
or say x & y can be a fraction value as well , insufficient

2.3^(x−1)(y−2)= 1
x,y not known ; insufficient

From1 & 2
x= 1 and y=2 we get 3^(x−1)(y−2)= 1
sufficient

“What is the value of xy?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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