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# If n is an integer, is (n+1)^2 an even integer?

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 7367
GMAT 1: 760 Q51 V42
GPA: 3.82
If n is an integer, is (n+1)^2 an even integer?  [#permalink]

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13 Sep 2018, 01:36
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84% (00:47) correct 16% (01:24) wrong based on 52 sessions

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[Math Revolution GMAT math practice question]

If $$n$$ is an integer, is $$(n+1)^2$$ an even integer?

1) $$n-1$$ is an even integer
2) $$(n-1)^2$$ is an even integer

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"Only $149 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Intern Joined: 02 May 2018 Posts: 11 GMAT 1: 620 Q46 V29 Re: If n is an integer, is (n+1)^2 an even integer? [#permalink] ### Show Tags 13 Sep 2018, 01:46 For $$(n+1)^2$$ to be even, $$n+1$$ should be even. A) $$n-1$$ is even implies that $$n+1$$ will be even and hence sufficient. B) $$(n-1)^2$$ is even implies that $$n-1$$ is even and thus $$n+1$$ will be even and hence sufficient. So answer is D VP Joined: 31 Oct 2013 Posts: 1354 Concentration: Accounting, Finance GPA: 3.68 WE: Analyst (Accounting) Re: If n is an integer, is (n+1)^2 an even integer? [#permalink] ### Show Tags 13 Sep 2018, 02:12 MathRevolution wrote: [Math Revolution GMAT math practice question] If $$n$$ is an integer, is $$(n+1)^2$$ an even integer? 1) $$n-1$$ is an even integer 2) $$(n-1)^2$$ is an even integer we are looking for whether $$(n + 1 )^2$$ is even or not. Statement 1: n - 1 = even. we know odd - odd = even. So, n is odd. So n +1 = even. Sufficient. Statement 2 : $$(n - 1 )^2$$= even. We know odd - odd = even and even times even is even. So, n is odd and n + 1 = even. Sufficient . Note: There is no effect of Exponents in odd even question . The best answer is D. CEO Joined: 12 Sep 2015 Posts: 3726 Location: Canada Re: If n is an integer, is (n+1)^2 an even integer? [#permalink] ### Show Tags 13 Sep 2018, 08:25 Top Contributor MathRevolution wrote: [Math Revolution GMAT math practice question] If $$n$$ is an integer, is $$(n+1)^2$$ an even integer? 1) $$n-1$$ is an even integer 2) $$(n-1)^2$$ is an even integer Target question: Is (n+1)² an even integer? This is a good candidate for rephrasing the target question. Aside: At the bottom of this point, you'll find a video with tips on rephrasing the target question (n+1)² = (n+1)(n+1). So, in order for (n+1)² to be even, it must be the case that n+1 is EVEN. Why is this? Well, if n+1 were ODD, then (n+1)² = (ODD)² = (ODD)(ODD) = ODD, but we want (n+1)² to be EVEN However, if n+1 were EVEN, then (n+1)² = (EVEN)² = (EVEN)(EVEN) = EVEN. Perfect. From here, we can see that if n+1 is EVEN, then it must be the case that n is ODD So, asking Is (n+1)² an even integer? is the same as asking Is n odd? REPHRASED target question: Is n odd? Statement 1: n-1 is an even integer If n-1 is an even integer, then we can be certain that n is odd Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT Statement 2: (n-1)² is an even integer If (n-1)² is an even integer, then we know that (n-1) is EVEN If (n-1) is EVEN, then we can be certain that n is odd Since we can answer the REPHRASED target question with certainty, statement 2 is SUFFICIENT Answer: D RELATED VIDEO FROM OUR COURSE _________________ Test confidently with gmatprepnow.com GMATH Teacher Status: GMATH founder Joined: 12 Oct 2010 Posts: 935 Re: If n is an integer, is (n+1)^2 an even integer? [#permalink] ### Show Tags 13 Sep 2018, 09:56 I would like to add some comments, related to dangerous situations usually explored in traps for the uncautious students... $${x^2}\,\,{\text{even}}\,\,\,\,{\text{does}}\,\,{\text{NOT}}\,\,{\text{imply}}\,\,\,\,\,x\,\,{\text{even}}\,\,\,\,\,\,\,{\text{ }}\left( {x = \sqrt 2 \,\,\,{\text{for}}\,\,{\text{example}}} \right)$$ $${y^2}\,\,{\text{odd}}\,\,\,\,{\text{does}}\,\,{\text{NOT}}\,\,{\text{imply}}\,\,\,\,\,y\,\,{\text{odd}}\,\,\,\,\,\,\,{\text{ }}\left( {y = \sqrt 3 \,\,\,{\text{for}}\,\,{\text{example}}} \right)$$ On the other hand, as it is the case in the problem proposed, $$x\,\,\,\operatorname{int} \,,\,\,\,{x^2}\,\,{\text{even}}\,\,\,\,{\text{imply}}\,\,\,\,\,\,x\,\,{\text{even}}\,\,\,\,\,\,\,{\text{ }}\left( {x\,\,{\text{is}}\,\,{\text{odd}}\,\,{\text{or}}\,\,{\text{even,}}\,\,{\text{but}}\,\,\,{\text{it}}\,\,{\text{is}}\,\,{\text{not}}\,\,{\text{odd}}...} \right)$$ $$y\,\,\,\operatorname{int} \,,\,\,\,{y^2}\,\,{\text{odd}}\,\,\,\,{\text{imply}}\,\,\,\,\,\,y\,\,{\text{odd}}\,\,\,\,\,\,\,{\text{ }}\left( {y\,\,{\text{is}}\,\,{\text{odd}}\,\,{\text{or}}\,\,{\text{even,}}\,\,{\text{but}}\,\,\,{\text{it}}\,\,{\text{is}}\,\,{\text{not}}\,\,{\text{even}}...} \right)$$ This kind of observation follows the notations and rationale taught in the GMATH method. Regards, Fabio. _________________ Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our high-level "quant" preparation starts here: https://gmath.net Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 7367 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If n is an integer, is (n+1)^2 an even integer? [#permalink] ### Show Tags 16 Sep 2018, 18:28 => 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. The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. $$(n+1)^2$$ is an even integer => $$n+1$$ is an even integer => $$n$$ is an odd integer Condition 1) Since “$$n-1$$ is an even integer” is equivalent to “$$n$$ is an odd integer”, condition 1) is sufficient. Condition 2) “$$(n-1)^2$$ is an even integer” is equivalent to “$$n-1$$ is an even integer”, which is condition 1). Condition 2) is also sufficient. Therefore, D is the answer. Answer: D Note: Tip 1) of the VA method states that D is most likely to be the answer if condition 1) gives the same information as condition 2). _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$149 for 3 month Online Course"
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Re: If n is an integer, is (n+1)^2 an even integer?   [#permalink] 16 Sep 2018, 18:28
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