MathRevolution wrote:
[GMAT math practice question]
(Number Properties) \(a\) and \(b\) are integers. If \(\frac{a}{504}\) is a terminating decimal, what is the value of \(a - b\)?
1) \(\frac{3}{b}\) is the simplest fraction of \(\frac{a}{504}\).
2) \(150 ≤ a ≤ 200\).
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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 \(\frac{a}{504}=\frac{a}{2^3∙3^2∙7}\) is a terminating decimal, \(a\) is a multiple of \(3^2 ·7 = 63.\)
Since we have \(2\) variables (\(a\) and \(b\)) and \(1\) equation, D 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) together tell us that we have \(ab = 3·504 = 2^3 · 3^3 · 7\) and \(a = 189.\)
Thus, we have \(b = \frac{(3·504)}{a} = \frac{(3·504)}{189} = \frac{504}{63} = 8.\)
Then we have \(a – b = 189 – 8 = 181\)
The answer is unique, and conditions 1) and 2) together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
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.
Let’s look at condition 1). It tells us that \(ab = 3·504 = 2^3 · 3^3 · 7.\)
If \(a = 189\) and \(b = 8\), then we have \(a – b = 189 – 8 = 181.\)
If \(a = 63\) and \(b = 24\), then we have \(a – b = 39.\)
The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.
Let’s look at the condition 2). It tells us that \(a = 189.\)
If \(a = 189\) and \(b = 8,\) then we have \(a – b = 189 – 8 = 181.\)
If \(a = 189, b = 1\), then we have \(a – b = 189 – 1 = 188.\)
The answer is not unique, and the condition is not 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.
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