MathRevolution wrote:
[GMAT math practice question]
(number properties) What is the value of \(x\)?
1) the prime factorization of \(x\) is \(ab(10a+b)\) (\(a, b\) are positive integers less than or equal to \(9\))
2) \(x\) is \(4\)-digit number
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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 \(3\) variables (\(x, a\) and \(b\)) and \(1\) equation, \(x = ab(10a+b)\), 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.
Conditions 1) and 2)
Since \(a\) and \(b\) are prime numbers less than or equal to \(9\) and \(10a + b\) is a prime number, the possible values of \(a\) and \(b\) are \(a = 2\) and \(b = 3; a = 3\) and \(b = 7; a = 5\) and \(b = 3;\) and \(a = 7\) and \(b = 3.\)
If \(a = 2\) and \(b = 3\), then \(x = 2*3*23 = 138,\) which is not a \(4\)-digit number.
If \(a = 3\) and \(b = 7,\) then \(x = 3*7*37 = 777,\) which is not a \(4\)-digit number.
If \(a = 5\) and \(b = 3\), then \(x = 5*3*53 = 795\), which is not a \(4\)-digit number.
If \(a = 7\) and \(b = 3,\) then \(x = 7*3*73 = 1533,\) which is a \(4\)-digit number.
\(x = 1533\) is the unique solution, and both conditions together are sufficient.
Since this question is a statistics 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 \(a\) and \(b\) are prime numbers less than or equal to \(9\) and \(10a + b\) is a prime number, the possible values of \(a\) and \(b\) are \(a = 2\) and \(b = 3; a = 3\) and \(b = 7; a = 5\) and \(b = 3;\) and \(a = 7\) and \(b = 3.\)
If \(a = 2\) and \(b = 3,\) then \(x = 2*3*23 = 138.\)
If \(a = 3\) and \(b = 7,\) then \(x = 3*7*37 = 777.\)
If \(a = 5\) and \(b = 3,\) then \(x = 5*3*53 = 795.\)
If \(a = 7\) and \(b = 3,\) then \(x = 7*3*73 = 1533.\)
Since condition 1) does not yield a unique solution, it is not sufficient.
Condition 2)
Condition 2) is obviously not sufficient on its own.
Therefore, C is the 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.