Two things required to answer..
1) whether \(X=y\)
2) value of n

1) xy=6..
X and y are 2 and 3...
We can say both are different..
But n not known
Insufficient
2) n=2..
X=y????!

If x=y..
\(x^ny^n=x^{2n}=x^{2*2}=x^4\)... Factors=4+1=5

X is different from y..
\(x^ny^n=x^2y^2\).. factors=(2+1)(2+1)=3*3=9

Insufficient

Combined...
X and y are different..
So factors=9
Sufficient

C
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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 3 variables (\(x, y\) and \(n\)) and 0 equations, E is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
Since \(x\) and \(y\) are prime numbers and \(xy = 6\), we must have \(x = 2\) and \(y = 3\), or \(x = 3\) and \(y = 2\).
If \(x = 2\) and \(y = 3\), then \(x^ny^n = 2^23^2\) has \((2+1)(2+1) = 9\) factors, since \(x\) and \(y\) are different prime numbers and \(n = 2\).
If \(x = 3\) and \(y = 2\), then \(x^ny^n = 3^22^2\) has \((2+1)(2+1) = 9\) factors, since \(x\) and \(y\) are different prime numbers and \(n = 2\).
Since we have a unique answer, both conditions together are sufficient.

Since this question is an integer question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
Since it doesn’t give us any information about the variable n, condition 1) is not sufficient.

Condition 2)
Since it doesn’t give us any information about the variables x and y, condition 2) is not sufficient.

Therefore, C is the answer.

Answer: C

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.
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