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If n is a positive integer, is

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by Max@Math Revolution » Wed Dec 12, 2018 1:17 am

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[Math Revolution GMAT math practice question]

If n is a positive integer, is
$$\sqrt{n+1}$$ an integer?

1) n is a multiple of 8
2) n is the product of 2 consecutive even numbers
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If n is a positive integer, is $$\sqrt{n+1}$$ an integer

by fskilnik@GMATH » Wed Dec 12, 2018 4:04 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If n is a positive integer, is
$$\sqrt{n+1}$$ an integer?

1) n is a multiple of 8
2) n is the product of 2 consecutive even numbers
$$n \ge 1\,\,{\mathop{\rm int}} \,\,\,\,\left( * \right)$$
$$\sqrt {n + 1} \,\,\mathop = \limits^? \,\,{\text{int}}\,\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\boxed{\,?\,\,\,\,:\,\,\,\,n + 1\,\,{\text{perfect}}\,\,{\text{square}}\,}$$

$$\left( 1 \right)\,\,n = 8M\,\,,\,\,M\mathop \ge \limits^{\left( * \right)} 1\,\,\,{\mathop{\rm int}} \,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,n = 8\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,n = 16\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.$$

$$\left( 2 \right)\,\,n + 1 = \left( {2K} \right)\left( {2K + 2} \right) + 1 = 4{K^2} + 4K + 1 = {\left( {2K + 1} \right)^2}\,,\,\,\,K\mathop \ge \limits^{\left( * \right)} 1\,\,\,{\mathop{\rm int}} \,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Max@Math Revolution » Thu Dec 13, 2018 11:42 pm
=>

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.

Condition 1)
If n = 8, then
$$\sqrt{n+1}=\sqrt{8+1}=\sqrt{9}=3$$
and the answer is 'yes'.
If n = 16, then
$$\sqrt{n+1}=\sqrt{16+1}=\sqrt{17}$$
and the answer is 'no'.
Thus, condition 1) is not sufficient since it does not yield a unique solution.

Condition 2)
If n is the product of two consecutive even integers, then
n = 2k(2k+2) = 4k^2 + 4k for some integer k.
$$\sqrt{n+1}=\sqrt{4k^2+4k+1}=\sqrt{\left(2k+1\right)^2}=2k+1$$
, and the answer is 'yes'.
Thus, condition 2) is sufficient.

Therefore, B is the answer.
Answer: B

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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