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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.
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Since we have 1 variable (x) and 0 equations, D is most likely the answer. So, we should consider each condition on its own first.
Let’s look at the condition 1). It tells us just the definition of a function f(x), therefore not giving us enough information to solve the question.
The answer is not unique, so the condition is not sufficient, according to Common Mistake Type 1, which states that the number of answers must be only one.
Let’s look at the condition 2). It tells us that f(420)·f(x) = 96. Since we do not have a definition of f(x), the answer is not unique, and condition 2) is not sufficient according to Common Mistake Type 1, which states that the number of answers must be only one.
Both conditions 1) & 2) together tell us that x has 4 factors for the following reason.
Remember the property that if n = p^aq^br^c where p, q, and r are different prime numbers, n has (a + 1)(b + 1)(c + 1) factors.
Since we have 420 = 2^23^15^17^1, it has (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 3·2·2·2 = 24 factors.
We have f(420)·f(x) = 24·f(x) = 96 or f(x) = 4, which means x has 4 factors.
Then we have two possibilities for x, which are x = p3 or x = p·q where p and q are different prime numbers.
Then we have 23 = 8 or 2·3 = 6 as the possible values of x.
Therefore, the minimum is 6.
The answer is unique, and both conditions 1) and 2) together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
Both conditions 1) & 2) together are sufficient.
Therefore, C is the correct answer.
Answer: C
If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations,” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A, B, C, or E.
