If m and n are positive integers and

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 2 variables (m and n) and 1 equation, D is most likely to be the answer. So, we should consider each of the conditions on its own first.

Condition 1)
There are at least two possible pairs of solutions. Two of these pairs are: m = 4, n = 9 and m = 9, n = 4.
Since we don’t have a unique solution, condition 1) is not sufficient.

Condition 2)
There are more than two pairs of solutions. Two of these are m = 9, n = 4 and m = 12, n = 3.
Since we don’t have a unique solution, condition 2) is not sufficient.

Conditions 1) and 2)
There are two possible pairs of solutions: m = 36, n = 1 and m = 9, n = 4.
Since we don’t have a unique solution, both conditions together are not sufficient.
Note: This question includes an example of a “hidden 1”.

Therefore, E is the answer.

Answer: E

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.

Step 1: Simplifying
√m√n=6
Squaring both the sides
mn = 36, that means m and n are the factors of 36


Understanding statement (1)
Possible ordered pair of m,n are (1,36), (4,9), (9,4) and (36,1)
n can be 1, 4, 9 36
Insufficient


As exact value of n cannot be calculated, statement is not sufficient

statement (1)+(2), possible ordered pair of m,n are (9,4) and (36,1)
n can still be 4 and 1
Insufficient

E is correct

Hello MathRevolution

This is a great Data sufficiency question.

Per the question, both m and n are positive integers
\(\sqrt{m}\sqrt{n}=6\)

Find n

Statement 1) Both \(m\), and \(n\) are the squares of integers.
So, \(\sqrt{m} and \sqrt{n} \) are integers.
Possible scenarios in this case are the following:
\(\sqrt{m} = 3\)
\(\sqrt{n} = 2 \)


\(\sqrt{m} = 2\)
\(\sqrt{n} = 3\)

\(\sqrt{m} = 6\)
\(\sqrt{n} = 1\)


\(\sqrt{m} = 1\)
\(\sqrt{n} = 6 \)

It is clear that the statement alone does not give sufficient information for evaluating value of n.
Therefore, Option A and D cannot be the answer.

Statement 2: \(m > n\)
This statement alone does not give any information sufficient to solve the question.
Therefore, Option B cannot be correct answer.

Combining Statement 1 & 2:
Possible scenarios are:
\(\sqrt{m} = 6 \)
\(\sqrt{n} = 1 \)

\(\sqrt{m} = 3 \)
\(\sqrt{n} = 2 \)

However, still no unique value of n can be ascertained.
Therefore, option C is not the correct answer.

This leaves us with Option E as the correct answer.

Hope this helps the interested students.
Is there anything anyone would like me to elaborate on? Pls share your thoughts on the solution. Thank you.