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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-------ASIDE----------------
If the prime factorization of N = (p^a)(q^b)(r^c) . . . (where p, q, r, etc are different prime numbers), then N has a total of (a+1)(b+1)(c+1)(etc) positive divisors.

Example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) =(5)(4)(2) = 40
-----NOW ONTO THE QUESTION----------------------

Target question: What is the number of factors of (x^n)(y^n)?

Statement 1: xy =6
There are several values of x, y and n that satisfy statement 1. Here are two:
Case a: x = 2, y = 3 and n = 1. We get the expression: (2^1)(3^1). So, the number of positive divisors = (1 + 1)(1 + 1) = 4. So, the answer to the target question is there are 4 divisors
Case b: x = 2, y = 3 and n = 7. We get the expression: (2^7)(3^7). So, the number of positive divisors = (7 + 1)(7 + 1) = 64. So, the answer to the target question is there are 64 divisors
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: n=2
There are several values of x, y and n that satisfy statement 2. Here are two:
Case a: x = 2, y = 3 and n = 2. We get the expression: (2^2)(3^2). So, the number of positive divisors = (2 + 1)(2 + 1) = 9. So, the answer to the target question is there are 9 divisors
Case b: x = 3, y = 3 and n = 2. We get the expression (3^2)(3^2), which simplifies to be (3^4). So, the number of positive divisors = (4 + 1) = 5. So, the answer to the target question is there are 5 divisors
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that one number is 2 and the other number is 3
Statement 2 tells us that n = 2
We get the expression: (2^2)(3^2). So, the number of positive divisors = (2 + 1)(2 + 1) = 9. So, the answer to the target question is there are 9 divisors
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent
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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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