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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 a three-digit integer, we have 3 variables, and E is most likely 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.
Conditions 1) & 2)
Since \(\sqrt{3x}\) is an integer, we have 3x = k^2 for some integer k.
Then k must be a multiple of 3, and we have k = 3n for some integer as well since 3 is a prime number.
Thus, we have 3x = k^2 = (3n)^2 = 9n^2 or x = 3n^2.
Since x is a 3-digit integer less than 200, we have 100 ≤ x = 3n^2 < 200 or 33.3 ≤ n^2 < 66.7 (dividing all terms by 3).
Then we have 34 ≤ n^2 ≤ 66 (because the answer is an integer according to the original condition) and squares between 34 and 66, inclusive are 36, 49, and 64.
The possible values of x are 3*36 = 108, 3*49 = 147 and 3*64 = 192.
Since both conditions together do not yield a unique solution, they are not sufficient.
Therefore, E is the answer.
Answer: E
In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.
