MathRevolution wrote:
[GMAT math practice question]
(Number Property) \(N\) is an integer. Is \(N\) a perfect square?
1) \(N\) is \(1\) greater than the product of \(4\) consecutive integers.
2) \(N\) is a summation of squares of \(4\) consecutive odd integers.
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Since we have \(1\) variable (\(N\)) and \(0\) equations, D is most likely the answer. So, we should consider each condition on its own first.
Condition 1)
Assume \(N\) is \(1\) greater than a product of four consecutive integers, \(x, x+1, x+2,\) and \(x+3\) where \(x\) is an integer.
We have
\(N = x(x + 1)(x + 2)(x + 3) + 1\)
\(N = x(x + 3)(x + 1)(x + 2) + 1 \)
\(N = (x^2 + 3x)(x^2 + 3x + 2) + 1 \)
\(N = (x^2 + 3x)^2 + 2(x^2 + 3x) + 1 \)
\(N = (x^2 + 3x + 1)^2\)
Thus, \(N\) is a perfect square.
Since condition 1) yields a unique solution, it is sufficient.
Condition 2)
If \(N = 1 + 3 + 5 + 7 = 16\), then \(N\) is a perfect square and the answer is ‘yes’.
If \(N = 3 + 5 + 7 + 9 = 24\), then \(N\) is not a perfect square and the answer is ‘no’.
Since condition 2) does not yield a unique solution, it is not sufficient.
Therefore, A is the answer.
Answer: A
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.