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N is an integer. Is N a perfect square?

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N is an integer. Is N a perfect square?

by Max@Math Revolution » Tue Mar 31, 2020 1:52 am

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[GMAT math practice question]

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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Source: — Data Sufficiency |
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Re: N is an integer. Is N a perfect square?

by Max@Math Revolution » Thu Apr 02, 2020 12:10 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.
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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.
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