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# If n is an integer, is n(n+2) divisible by 8?

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If n is an integer, is n(n+2) divisible by 8?  [#permalink]

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19 Sep 2018, 02:01
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[Math Revolution GMAT math practice question]

If $$n$$ is an integer, is $$n(n+2)$$ divisible by $$8$$?

1) $$n$$ is an even number.
2) $$n$$ is a multiple of $$4$$.

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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself"  Math Revolution Discount Codes e-GMAT Discount Codes Veritas Prep GMAT Discount Codes Senior Manager Joined: 18 Jul 2018 Posts: 274 Location: India Concentration: Finance, Marketing WE: Engineering (Energy and Utilities) Re: If n is an integer, is n(n+2) divisible by 8? [#permalink] ### Show Tags 19 Sep 2018, 03:06 From statement 1: n is even. The least even number is zero. n=0 n(n+2) = 0. Which is divisible by 8. 1 is sufficient. From statement 2: n is a multiple of 4. The least value of n is 4. n=4 n(n+2) = 8. Which is divisible by 8. 2 is also sufficient. D is the answer. _________________ When you want something, the whole universe conspires in helping you achieve it. CEO Joined: 12 Sep 2015 Posts: 3020 Location: Canada Re: If n is an integer, is n(n+2) divisible by 8? [#permalink] ### Show Tags 19 Sep 2018, 06:58 Top Contributor MathRevolution wrote: [Math Revolution GMAT math practice question] If $$n$$ is an integer, is $$n(n+2)$$ divisible by $$8$$? 1) $$n$$ is an even number. 2) $$n$$ is a multiple of $$4$$. Target question: Is n(n+2) divisible by 8? Statement 1: n is an even number. Let's examine some CONSECUTIVE even numbers: 6, 8, 10, 12, 14, 16, 18, 20, 22, etc Notice that the numbers increase by 2 with each subsequent value. So, if n is EVEN, then we know that n+2 is also even. Also, notice that for any pair of CONSECUTIVE even integers, one value is divisible by 2 and the other is divisible by 4. That is, for any pair of CONSECUTIVE even integers, one value can be written as (2)(some integer) and the other value can be written as (4)(some integer) In other words, we can write: n(n+2) = (2)(some integer)(4)(some integer) = (8)(some integer) This means, n(n+2) IS divisible by 8 Since we can answer the target question with certainty, statement 1 is SUFFICIENT Statement 2: n is a multiple of 4 This tells is that n is even. So, n and n+2 are CONSECUTIVE even numbers This means we can apply the exact same logic we applied with statement 1 and conclude that n(n+2) IS divisible by 8 Since we can answer the target question with certainty, statement 2 is SUFFICIENT Answer: D Cheers, Brent _________________ Brent Hanneson – GMATPrepNow.com Sign up for our free Question of the Day emails Rice (Jones) Thread Master Joined: 18 Jun 2018 Posts: 66 Location: United States (AZ) Concentration: Finance, Healthcare If n is an integer, is n(n+2) divisible by 8? [#permalink] ### Show Tags 19 Sep 2018, 07:12 Zero is also a multiple of 4. No? Afc0892 Afc0892 wrote: From statement 1: n is even. The least even number is zero. n=0 n(n+2) = 0. Which is divisible by 8. 1 is sufficient. From statement 2: n is a multiple of 4. The least value of n is 4. n=4 n(n+2) = 8. Which is divisible by 8. 2 is also sufficient. D is the answer. Intern Joined: 19 Jun 2016 Posts: 19 Location: India Concentration: Strategy GPA: 3 Re: If n is an integer, is n(n+2) divisible by 8? [#permalink] ### Show Tags 19 Sep 2018, 09:45 I did not understand. What if n=-2, then n(n+2) becomes 0, which is not divisible by 8. Am I missing something here? Thanks. Posted from my mobile device Rice (Jones) Thread Master Joined: 18 Jun 2018 Posts: 66 Location: United States (AZ) Concentration: Finance, Healthcare Re: If n is an integer, is n(n+2) divisible by 8? [#permalink] ### Show Tags 19 Sep 2018, 09:48 bharu130 Zero is divisible b 8 since zero is a multiple of all numbers. Hope this helps. Cheers! bharu130 wrote: I did not understand. What if n=-2, then n(n+2) becomes 0, which is not divisible by 8. Am I missing something here? Thanks. Posted from my mobile device Sent from my iPhone using GMAT Club Forum mobile app Senior Manager Joined: 18 Jul 2018 Posts: 274 Location: India Concentration: Finance, Marketing WE: Engineering (Energy and Utilities) Re: If n is an integer, is n(n+2) divisible by 8? [#permalink] ### Show Tags 19 Sep 2018, 18:20 funsogu wrote: Zero is also a multiple of 4. No? Afc0892 Afc0892 wrote: From statement 1: n is even. The least even number is zero. n=0 n(n+2) = 0. Which is divisible by 8. 1 is sufficient. From statement 2: n is a multiple of 4. The least value of n is 4. n=4 n(n+2) = 8. Which is divisible by 8. 2 is also sufficient. D is the answer. Yes, zero is a multiple of every number. Posted from my mobile device _________________ When you want something, the whole universe conspires in helping you achieve it. GMATH Teacher Status: GMATH founder Joined: 12 Oct 2010 Posts: 379 Re: If n is an integer, is n(n+2) divisible by 8? [#permalink] ### Show Tags 19 Sep 2018, 18:49 MathRevolution wrote: If $$n$$ is an integer, is $$n(n+2)$$ divisible by $$8$$? 1) $$n$$ is an even number. 2) $$n$$ is a multiple of $$4$$. $$n\,\,{\mathop{\rm int}}$$ $${{n\left( {n + 2} \right)} \over 8}\,\,\mathop = \limits^? \,\,{\mathop{\rm int}}$$ $$\left( 1 \right)\,\,n = 2M,\,\,M\,\,{\mathop{\rm int}}$$ $${{n\left( {n + 2} \right)} \over 8} = {{2M \cdot 2 \cdot \left( {M + 1} \right)} \over 8} = {{M\left( {M + 1} \right)} \over 2}\mathop = \limits^{\left( * \right)} \,\,{\mathop{\rm int}} \,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.$$ (*) The product of two consecutive integers is even (odd*even or even*odd), hence the integer M(M+1) has (at least) one factor 2 in its decomposition... $$\left( 2 \right)\,\,n = 4J\,\,,\,\,J\,\,{\mathop{\rm int}}$$ $${{n\left( {n + 2} \right)} \over 8} = {{4J \cdot 2 \cdot \left( {2J + 1} \right)} \over 8} = J \cdot \left( {2J + 1} \right) = {\mathop{\rm int}} \cdot {\mathop{\rm int}} = {\mathop{\rm int}} \,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.$$ This solution follows the notations and rationale taught in the GMATH method. Regards, fskilnik. _________________ Fabio Skilnik :: http://www.GMATH.net (Math for the GMAT) Course release PROMO : finish our test drive till 31/Oct with (at least) 60 correct answers out of 92 (12-questions Mock included) to gain a 60% discount! Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6390 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If n is an integer, is n(n+2) divisible by 8? [#permalink] ### Show Tags 21 Sep 2018, 01:22 => 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. Since we have $$1$$ variable (n) and $$0$$ equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first. Condition 1) If $$n$$ is an even integer, then $$n$$ and $$n + 2$$ are two consecutive even integers. Products of two consecutive even integers are multiples of $$8$$ since one of them must be a multiple of $$4$$, and the other a multiple of $$2$$. Condition 1) is sufficient. Condition 2) Since $$n$$ is a multiple of $$4, n + 2$$ is an even integer. Thus, $$n(n+2)$$ is a multiple of $$8$$. Condition 2) is sufficient. Therefore, D is the answer. Answer: D 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. _________________ 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$99 for 3 month Online Course"
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Re: If n is an integer, is n(n+2) divisible by 8? &nbs [#permalink] 21 Sep 2018, 01:22
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# If n is an integer, is n(n+2) divisible by 8?

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