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x in an integer. Is x a perfect square?

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x in an integer. Is x a perfect square?

by Max@Math Revolution » Tue Apr 21, 2020 1:26 am

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[GMAT math practice question]

x in an integer. Is x a perfect square?

1) x is one greater than the product of 4 consecutive positive integers.
2) x is the sum of five consecutive odd numbers.
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Source: — Data Sufficiency |
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Re: x in an integer. Is x a perfect square?

by Max@Math Revolution » Thu Apr 23, 2020 2:07 am

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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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Since we have 1 variable (x) and 0 equations, D is most likely the answer. So, we should consider each condition on its own first.

Condition 1)

Assume x = k(k + 1)(k + 2)(k + 3) + 1.
Then x = (k^2 + 3k)(k^2 + 3k + 2) + 1 = A(A + 2) + 1 = A^2 + 2A + 1 = (A + 1)^2 for A = k^2 + 3k for an integer k.

Thus, x is a perfect integer, and the answer is ‘yes’.

Since condition 1) yields a unique solution, it is sufficient.

Condition 2)

If x = 1 + 3 + 5 + 7 + 9 = 25, then x is a perfect integer and the answer is ‘yes’.
If x = 3 + 5 + 7 + 9 + 11 = 35, then x is not a perfect integer 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.
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