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If p and q are prime numbers, what is the number of factors

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by Max@Math Revolution » Fri Apr 12, 2019 12:28 am

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[GMAT math practice question]

If p and q are prime numbers, what is the number of factors of 6pq?

1) p and q are different
2) p < 3 < q
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Source: — Data Sufficiency |
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by Max@Math Revolution » Sun Apr 14, 2019 5:29 pm
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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.

Recall that if n = p^aq^br^c, where p, q and r are different prime numbers, and a, b and c are non-negative integers, then n has (a+1)(b+1)(c+1) factors.

Condition 2)
We must have p = 2 since p is prime and p < 3.
The prime factorization of 6pq is 2*3*p*q = 2^2*3*q since q is prime and q > 3.
The number of factors of 2^2*3*q is (2+1)(1+1)(1+1) = 12. Condition 2) is sufficient since it yields a unique answer.

Condition 1)
If p = 2 and q = 3, then 6pq = 2^2*3^2 and the number of factors is (2+1)(2+1)= 9.
If p = 5 and q = 7, then 6pq = 2*3*5*7 and the number of factors is (1+1)(1+1)(1+1)(1+1) = 8.
Condition 1) is not sufficient since it does not yield a unique solution.

Therefore, B is the answer.
Answer: B
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