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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 4 variables (\(p,q,m\), and \(n\)) and 0 equations, the answer is most likely to be E.

As E is the most likely answer, we should consider both conditions 1) and 2) together before considering each of them individually. If they are not sufficient when taken together, E is the answer.

Conditions 1) & 2)

The two conditions yield \(p^mq^n = 11^2x3^3\).

Since \(11\) and \(13\) are different prime numbers, the number of factors is \((2+1)(3+1) = 12\).

Both conditions are sufficient, when taken together.

Since this is an integer question (one of the key question areas), we should also consider choices A and B by CMT 4(A).

Condition 1)

This condition does not tell us whether \(p=q\). Therefore, it is not sufficient.

Condition 2)

Since this condition does not tell us the values of the exponents, we can’t determine the number of factors. This condition is not sufficient

Note: It is important to look out for the word, “different” in factor or prime factor questions.

For example, suppose we are told that \(m\) and \(n\) are positive integers, and that \(p\) and \(q\) are different prime numbers. If we are then asked how many factors \(p^mq^n\) has, and given the conditions

1) \(m=2\) and \(n=3\)

2) \(p=11\) and \(q=13\),

the answer will be A. As we know that the number of factors of \(p^mq^n\) is \((m+1)(n+1)\), the information provided by Condition 1) is sufficient (Condition 2) gives us no information about the exponents, and so is not sufficient).

Answer: C

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