MathRevolution wrote:
Que: How many distinct positive factors do the integer m have?
(1) m = \(p^3 q^2\), where p and q are distinct positive prime numbers.
(2) The only positive prime factors of m are 2 and 3.
Solution: To save time and improve accuracy on DS questions in GMAT, learn and apply the Variable Approach.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.
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Now we will solve this DS question using the Variable Approach.
Let’s apply the 3 steps suggested previously.
Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.
We have to find the number of distinct positive factors of the integer m
Follow the second and the third step: From the original condition, we have 1 variable (m). To match the number of variables with the number of equations, we need 1 equation. Since conditions (1) and (2) will provide 1 equation each, D would most likely be the answer.
Recall 3- Principles and Choose D as the most likely answer. Let’s look at each condition separately.
Condition (1) tells us that m = \(p^3 q^2\), where p and q are distinct positive prime numbers.
The number of positive factors of a number N, expressed in its prime form as \(N = p^x q^y\), where p and q are distinct primes, is given by (x + 1) (y + 1).
=> m = \(p^3 q^2\). Thus, the number of positive factors = (3 + 1) (2 + 1) = 12
The answer is unique; condition (1) alone is sufficient according to Common Mistake Type 2 which states that the answer should be a unique value.
Condition (2) tells us that the only positive prime factors of m are 2 and 3.
Here, m can take multiple possible values. For example, 2 × 3, \(2^2 × 3^3\), \(2^4 × 3\), etc. all have 2 and 3 as the only two prime factors. However, the number of factors of each of the above numbers is different.
The answer is not a unique value; condition (2) alone is not sufficient according to Common Mistake Type 2 which states that the answer should be a unique value.
Condition (1) alone is not sufficient.
Therefore, A is the correct answer.
Answer: A _________________