x and y are positive integers. If p and q are diff

[Max@Math Revolution]

[GMAT math practice question]

x and y are positive integers. If p and q are different prime numbers, what is the number of factors of p^xq^y?

1) x=2 and y=3
2) p=2 and q=3

[GMATGuruNY]

To determine the number of positive factors of an integer:

1) Prime-factorize the integer
2) Add 1 to each exponent
3) Multiply

For example:
72 = 2³ * 3².
Adding 1 to each exponent and multiplying, we get (3+1)*(2+1) = 12 factors.

Here's why:
To determine how many factors can be created from 72 = 2³ * 3², we need to determine the number of choices we have of each prime factor and to count the number of ways these choices can be combined:

For 2, we can use 2�, 2¹, 2², or 2³, giving us 4 choices.
For 3, we can use 3�, 3¹, or 3², giving us 3 choices.

Multiplying the number of choices we have of each factor, we get 4*3 = 12 possible factors.

[GMAT math practice question]

x and y are positive integers. If p and q are different prime numbers, what is the number of factors of p^xq^y?

1) x=2 and y=3
2) p=2 and q=3

In accordance with the explanation above, the number of factors for the expression in blue is equal to the following:
(x+1)(y+1).
Question stem, rephrased:
What is the value of (x+1)(y+1)?

Statement 1:
(x+1)(y+1) = (2+1)(3+1) = 12.
SUFFICIENT.

Statement 2:
No way to determine the value of (x+1)(y+1).
INSUFFICIENT.

The correct answer is A.

[Max@Math Revolution]

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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.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

Since p and q are different prime numbers, the number of factors of p^xq^y is (x+1)(y+1).
Thus, the question asks for the value of (x+1)(y+1) and only condition 1) is sufficient.

Therefore, A is the answer.

Answer: A

When we count the number of factors, the prime numbers must be different in prime number factorizations.