Bunuel wrote:
If x and y are both integers greater than 1, is xy > 100?
(1) x has exactly 7 unique factors.
(2) y has exactly 9 unique factors.
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 2 variables (\(x\) and \(y\)) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.
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Conditions 1) & 2)
Since the integer x is \(p^6\) type of integer greater than 1 by condition 1), where \(p\) is a prime integer, the smallest possible number of is \(2^6 = 64\).
Since the integer y is \(p^8\) or \(p^2*q^2\) types of integers greater than 1 condition 2), where \(p\) and \(q\) are prime numbers, the smallest possible number of y is \(2^2*3^2 = 36\).
Then, the smallest possible value of \(xy\) is \(64*36 > 100\) and the answer is 'yes'.
Thus, both conditions together are sufficient.
Since this question is an integer question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.
Condition 1)
Since the integer x is \(p^6\) type of integer greater than 1 by condition 1), where \(p\) is a prime integer, the smallest possible number of is \(2^6 = 64\).
Since the smallest possible number of \(y\) is 2, the smallest possible number of \(xy\) is 128, which is greater than 100 and the answer is yes.
That's why condition 1) is sufficient.
Condition 2)
Since the integer y is \(p^8\) or \(p^2*q^2\) types of integers greater than 1 condition 2), where \(p\) and \(q\) are prime numbers, the smallest possible number of y is \(2^2*3^2 = 36\).
If \(x = 2\) and \(y = 36\), then \(xy = 72\) is less than 100 and the answer is "no".
If \(x = 3\) and \(y = 36\), then \(xy = 108\) is greater than 100 and the answer is "yes".
Since condition 2) doesn't yield a unique answer, it is not sufficient.
Therefore, A is the answer.
Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
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