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# If k is a positive integer and n = k(k + 7k), is n divisible

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Joined: 28 May 2013
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Schools: Duke '17 (WA$) WE: Military Officer (Military & Defense) If k is a positive integer and n = k(k + 7k), is n divisible [#permalink] ### Show Tags Updated on: 10 Apr 2014, 06:42 8 74 00:00 Difficulty: 55% (hard) Question Stats: 65% (02:04) correct 35% (02:14) wrong based on 1121 sessions ### HideShow timer Statistics If k is a positive integer and n = k(k + 7), is n divisible by 6? (1) k is odd. (2) When k is divided by 3, the remainder is 2. Originally posted by Jem2905 on 02 Nov 2013, 14:01. Last edited by AbhiJ on 10 Apr 2014, 06:42, edited 2 times in total. Renamed the topic, edited the question and added the OA. ##### Most Helpful Expert Reply Math Expert Joined: 02 Sep 2009 Posts: 58452 Re: If k is a positive integer and n = k(k+7)..... [#permalink] ### Show Tags 10 Nov 2013, 12:30 26 20 ashsim wrote: Chiranjeevee wrote: Jem2905 wrote: Hi guys, trying to get a little help on this problem that stumped me recently on a practice test. After going back and spending some more time with it, I got a different answer but I'm not sure if it's the right answer, and I'm not exactly sure I understand why it's the correct answer... any resphrasing of the question or statements will be hugely appreciated. Thanks!! If k is a positive integer and n = k(k + 7k), is n divisible by 6? (1) k is odd. (2) When k is divided by 3, the remainder is 2. Given, n= k(k+7K) = 8k^2 now for n to be divisible by 6, k should be divisible by 3. 1. K is odd. clearly insufficient, for k=1 answer is No. for k=3, answer is yes. 2. When k is divided by 3, the remainder is 2. Remainder is 2, so K can never be divisible by 3, Hence n will not be divisible by 6. So Sufficient. IMO, B Hi, I'm trying to understand this question too. The question I saw had n=K(K+7), not k+ 7K as written in the question above. Can anyone explain this q with the change please? Thanks! If k is a positive integer and n = k(k + 7), is n divisible by 6? (1) k is odd. If $$k = 1$$, then $$n = k(k + 7) = 8$$ and n is NOT divisible by 6 but if $$k = 3$$, then $$n = k(k + 7) = 30$$ and n IS divisible by 6. Not sufficient. (2) When k is divided by 3, the remainder is 2 --> $$k = 3x + 2$$ --> $$n = k(k + 7) = (3x + 2)(3x + 9)=9x^2+33 x+18=3(3x^2+11x)+18$$. Notice that $$3x^2+11x$$ is even no matter whether x is even or odd, thus $$n=3(3x^2+11x)+18=3*even+(a \ multiple \ of \ 6)=(a \ multiple \ of \ 6)+(a \ multiple \ of \ 6)=(a \ multiple \ of \ 6)$$. Sufficient. Answer: B. Hope it's clear. _________________ ##### Most Helpful Community Reply Manager Joined: 15 Aug 2013 Posts: 50 Re: If k is a positive integer and n = k(k+7)..... [#permalink] ### Show Tags 11 Nov 2013, 03:23 22 2 Here is my solution- If n=K(K+7), then- 1) k is odd => odd * (odd + Even) = Odd * Odd. We can not say it is multiple of 6 or not. Insufficient. 2) k = 3m +2. So, n=K(K+7) => n= (3m+2)(3m+9)= 3(3m+2)(m+3) => multiple of 3. If m is odd, m+3 is even. Hence , multiple of 2. Also it is a multiple of 3 => multiple of 6 If m is even, 3m+2 is even. Hence , multiple of 2. Also it is a multiple of 3 => multiple of 6 So B is sufficient. ##### General Discussion Intern Joined: 28 Jan 2013 Posts: 28 Re: If k is a positive integer and n = k(k+7)..... [#permalink] ### Show Tags 02 Nov 2013, 22:08 2 3 Jem2905 wrote: Hi guys, trying to get a little help on this problem that stumped me recently on a practice test. After going back and spending some more time with it, I got a different answer but I'm not sure if it's the right answer, and I'm not exactly sure I understand why it's the correct answer... any resphrasing of the question or statements will be hugely appreciated. Thanks!! If k is a positive integer and n = k(k + 7k), is n divisible by 6? (1) k is odd. (2) When k is divided by 3, the remainder is 2. Given, n= k(k+7K) = 8k^2 now for n to be divisible by 6, k should be divisible by 3. 1. K is odd. clearly insufficient, for k=1 answer is No. for k=3, answer is yes. 2. When k is divided by 3, the remainder is 2. Remainder is 2, so K can never be divisible by 3, Hence n will not be divisible by 6. So Sufficient. IMO, B Intern Joined: 04 Nov 2013 Posts: 1 Re: If k is a positive integer and n = k(k+7)..... [#permalink] ### Show Tags 04 Nov 2013, 17:51 6 What about 5? It is a positive integer, when divided by 3 the remainder is 2 and 5(5+7)= 5(12) = 60, divisible by 6. Intern Joined: 09 Nov 2013 Posts: 9 Re: If k is a positive integer and n = k(k+7)..... [#permalink] ### Show Tags 10 Nov 2013, 08:45 Chiranjeevee wrote: Jem2905 wrote: Hi guys, trying to get a little help on this problem that stumped me recently on a practice test. After going back and spending some more time with it, I got a different answer but I'm not sure if it's the right answer, and I'm not exactly sure I understand why it's the correct answer... any resphrasing of the question or statements will be hugely appreciated. Thanks!! If k is a positive integer and n = k(k + 7k), is n divisible by 6? (1) k is odd. (2) When k is divided by 3, the remainder is 2. Given, n= k(k+7K) = 8k^2 now for n to be divisible by 6, k should be divisible by 3. 1. K is odd. clearly insufficient, for k=1 answer is No. for k=3, answer is yes. 2. When k is divided by 3, the remainder is 2. Remainder is 2, so K can never be divisible by 3, Hence n will not be divisible by 6. So Sufficient. IMO, B Hi, I'm trying to understand this question too. The question I saw had n=K(K+7), not k+ 7K as written in the question above. Can anyone explain this q with the change please? Thanks! Intern Joined: 27 Jun 2013 Posts: 3 Re: If k is a positive integer and n = k(k+7)..... [#permalink] ### Show Tags 07 Jan 2014, 15:01 Hey guys, I still have a question on this problem, can someone explain why my solution is incorrect? If k is a positive integer and n = k(k + 7), is n divisible by 6? 1. K is odd Test cases: K=1 n=1(1+7) = 8, is 8/6 NO K=3 n=3(3+7)=30, is 30/6 YES INSUFFICIENT 2. When k is divided by 3, the remainder is 2 Test Cases: K=1 1/3=0 remainder 2, n=1(1+7) = 8, is 8/6 NO K=5 5/3 =1 remainder 2, n=5(5+7) =60, is 60/6 YES INSUFFICIENT Combined: K=1 overlaps - NO K=5 overlaps - YES I see the math approach in the posts above but why would the test cases produce a different result? What am I missing here? Thanks in advance. Math Expert Joined: 02 Sep 2009 Posts: 58452 Re: If k is a positive integer and n = k(k+7)..... [#permalink] ### Show Tags 08 Jan 2014, 03:49 1 msbandi4321 wrote: Hey guys, I still have a question on this problem, can someone explain why my solution is incorrect? If k is a positive integer and n = k(k + 7), is n divisible by 6? 1. K is odd Test cases: K=1 n=1(1+7) = 8, is 8/6 NO K=3 n=3(3+7)=30, is 30/6 YES INSUFFICIENT 2. When k is divided by 3, the remainder is 2 Test Cases: K=1 1/3=0 remainder 2, n=1(1+7) = 8, is 8/6 NO K=5 5/3 =1 remainder 2, n=5(5+7) =60, is 60/6 YES INSUFFICIENT Combined: K=1 overlaps - NO K=5 overlaps - YES I see the math approach in the posts above but why would the test cases produce a different result? What am I missing here? Thanks in advance. 1 divided by 3 yields the remainder of 1, not 2: 1=0*3+1. Does this make sense? _________________ Intern Joined: 19 Nov 2013 Posts: 13 Schools: Haas '17 (S) GMAT 1: 680 Q47 V36 Re: If k is a positive integer and n = k(k + 7k), is n divisible [#permalink] ### Show Tags 09 Apr 2014, 16:51 I still don't see why Statement 2 is Sufficient. If you use 5, the outcome is 60 (divisible by 60), and if you use 8 the outcome is 320 (not divisible by 6). Please explain. Math Expert Joined: 02 Sep 2009 Posts: 58452 Re: If k is a positive integer and n = k(k + 7k), is n divisible [#permalink] ### Show Tags 10 Apr 2014, 02:22 jbartuccio wrote: I still don't see why Statement 2 is Sufficient. If you use 5, the outcome is 60 (divisible by 60), and if you use 8 the outcome is 320 (not divisible by 6). Please explain. If k=8, then n=k(k+7)=8*15=120, not 320 and 120 is divisible by 6. The reason why the second statement is sufficient is given here: if-k-is-a-positive-integer-and-n-k-k-7k-is-n-divisible-162594.html#p1290699 Does it make sense? _________________ Intern Joined: 08 Apr 2014 Posts: 12 Re: If k is a positive integer and n = k(k + 7k), is n divisible [#permalink] ### Show Tags 10 Apr 2014, 02:58 jbartuccio wrote: I still don't see why Statement 2 is Sufficient. If you use 5, the outcome is 60 (divisible by 60), and if you use 8 the outcome is 320 (not divisible by 6). Please explain. if you refer to the initial question: n= k(k+7k) = 8k^2 Hence, if k = 5, n = 8*5*5 if k = 8, n = 8*8*8 both of them are clearly not divisible by 6. *press kudos if you like the answer Intern Joined: 08 Jan 2015 Posts: 2 Re: If k is a positive integer and n = k(k + 7k), is n divisible [#permalink] ### Show Tags 02 Feb 2015, 19:16 3 1. K is odd It's clearly insufficient. Test k=1 and k=3. 2. When k is divided by 3, the remainder is 2 n=k(k+7)=k(k+1+6) --> n=k(k+1)+6k 6k is always divisible by 6. Need to show that k(k+1) is also divisible by 6. When k is divided by 3, the remainder is 2, then k=2,5,8,11,... for k=2,5,8,11,..., k(k+1) is always divisible by 6: k=2: k(k+1)=2*3=6 k=5: 5*6 k=8: 8*9 k=11: 11*12 k(k+1) and 6k are both divisible by 6, therefore n=k(k+1)+6k divisible by 6. sufficient The answer is B. Manager Joined: 24 Dec 2014 Posts: 128 Location: India Concentration: International Business, Finance GMAT 1: 700 Q47 V39 GMAT 2: 790 Q51 V50 GPA: 3.65 WE: Operations (Energy and Utilities) Re: If k is a positive integer and n = k(k + 7k), is n divisible [#permalink] ### Show Tags 02 Feb 2015, 19:38 ricsingh wrote: jbartuccio wrote: I still don't see why Statement 2 is Sufficient. If you use 5, the outcome is 60 (divisible by 60), and if you use 8 the outcome is 320 (not divisible by 6). Please explain. if you refer to the initial question: n= k(k+7k) = 8k^2 Hence, if k = 5, n = 8*5*5 if k = 8, n = 8*8*8 both of them are clearly not divisible by 6. *press kudos if you like the answer Hi ricsingh: n=k(k+7) is not same as n=k(k+7k). Watch out for silly mistakes! Posted from my mobile device Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 9706 Location: Pune, India Re: If k is a positive integer and n = k(k + 7k), is n divisible [#permalink] ### Show Tags 02 Feb 2015, 22:29 6 3 Jem2905 wrote: If k is a positive integer and n = k(k + 7), is n divisible by 6? (1) k is odd. (2) When k is divided by 3, the remainder is 2. Given: n = k(k + 7) Question: Is n divisible by 6? (1) k is odd. If k = 1, n = 8 - Not divisible by 6 If k = 6, n is divisible by 6 Not sufficient (2) When k is divided by 3, the remainder is 2. k = (3b+2) n = (3b+2)(3b+2 + 7) = (3b + 2)(3b + 9) = 3*(3b + 2)(b + 3) For n to be divisible by 6, it must be divisible by both 2 and 3. We see that it is divisible by 3. Let's see if it is divisible by 2 too i.e. if it is even. b can be odd or even in this expression. If it is odd, (b+3) will become even because (Odd + Odd = Even). If it is even, (3b+2) will become even because (Even + Even = Even). So in either case, n will be even. So n will be divisible by 3 as well as 2 i.e. it will be divisible by 6. Sufficient alone. Answer (B) _________________ Karishma Veritas Prep GMAT Instructor Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options > Manager Joined: 08 Oct 2013 Posts: 50 Re: If k is a positive integer and n = k(k + 7k), is n divisible [#permalink] ### Show Tags 05 May 2015, 11:03 n= k(k+7) so is n divisible by 6? State 1: K is Odd Possible values are 1, 3, 5, 7 etc Expression is not true if n = 1 but true for rest of the numbers so Insufficient State 2: When K is divided by 3, the remainder is 2. Possible values are 2, 8, 11, 14, 17 etc For all there values k(k+7) is divisible by 6 - Hence State 2 is Sufficient. Answer is B Intern Joined: 08 Oct 2015 Posts: 7 Re: If k is a positive integer and n = k(k + 7k), is n divisible [#permalink] ### Show Tags 10 Nov 2015, 22:22 Bunuel wrote: If k is a positive integer and n = k(k + 7), is n divisible by 6? (1) k is odd. If $$k = 1$$, then $$n = k(k + 7) = 8$$ and n is NOT divisible by 6 but if $$k = 3$$, then $$n = k(k + 7) = 30$$ and n IS divisible by 6. Not sufficient. (2) When k is divided by 3, the remainder is 2 --> $$k = 3x + 2$$ --> $$n = k(k + 7) = (3x + 2)(3x + 9)=9x^2+33 x+18=3(3x^2+11x)+18$$. Notice that $$3x^2+11x$$ is even no matter whether x is even or odd, thus $$n=3(3x^2+11x)+18=3*even+(a \ multiple \ of \ 6)=(a \ multiple \ of \ 6)+(a \ multiple \ of \ 6)=(a \ multiple \ of \ 6)$$. Sufficient. Answer: B. Hope it's clear. Bunuel, Can you explain why : $$3x^2+11x$$ is even no matter whether x is even or odd? I'm sure there is a simple theoretical way to see this quicker than plugging in numbers. Thanks! Math Expert Joined: 02 Sep 2009 Posts: 58452 Re: If k is a positive integer and n = k(k + 7k), is n divisible [#permalink] ### Show Tags 10 Nov 2015, 23:35 dubyap wrote: Bunuel wrote: If k is a positive integer and n = k(k + 7), is n divisible by 6? (1) k is odd. If $$k = 1$$, then $$n = k(k + 7) = 8$$ and n is NOT divisible by 6 but if $$k = 3$$, then $$n = k(k + 7) = 30$$ and n IS divisible by 6. Not sufficient. (2) When k is divided by 3, the remainder is 2 --> $$k = 3x + 2$$ --> $$n = k(k + 7) = (3x + 2)(3x + 9)=9x^2+33 x+18=3(3x^2+11x)+18$$. Notice that $$3x^2+11x$$ is even no matter whether x is even or odd, thus $$n=3(3x^2+11x)+18=3*even+(a \ multiple \ of \ 6)=(a \ multiple \ of \ 6)+(a \ multiple \ of \ 6)=(a \ multiple \ of \ 6)$$. Sufficient. Answer: B. Hope it's clear. Bunuel, Can you explain why : $$3x^2+11x$$ is even no matter whether x is even or odd? I'm sure there is a simple theoretical way to see this quicker than plugging in numbers. Thanks! $$3x^2+11x=x(3x+11)$$. If x is even the result is obviously even: $$x(3x+11)=even*integer=even$$; If x is odd, then $$x(3x+11)=odd(odd*odd+odd)=odd*even=even$$. Hope it's clear. _________________ Intern Joined: 21 Jun 2014 Posts: 28 If k is a positive integer and n = k(k + 7k), is n divisible [#permalink] ### Show Tags 11 Nov 2015, 08:30 1 Jem2905 wrote: If k is a positive integer and n = k(k + 7), is n divisible by 6? (1) k is odd. (2) When k is divided by 3, the remainder is 2. Divisibility by 6 means must be divisible by 3x2 (Prime Factors of 6) 1) k is Odd ==> k(k+7) = Odd(Odd+Odd) = Even This will give me at least one factor of 2.. but nothing about factor of 3 [Insufficient] 2) k Divided by 3 ==> Remainder = 2 Now Possible values could be 2,5,8,11 -- Alternatively Even and Odd So if k=2,8,... Then I am getting One factor of 2 and when I add these to 7 I get Multiple of 3, Similarly when I I choose k=5,11.. Odd Number with remainder of 2 and I add them to 7 I get Even Multiple of 3 i.e. Multiple of 6 This is happening because, Numbers with Remainders of 2 when added to 7, which leaves remainder of 1 when divided by 3, makes the Sum Divisible by 3 [Sufficient] <--- Answer is B Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8017 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If k is a positive integer and n = k(k + 7k), is n divisible [#permalink] ### Show Tags 15 Nov 2015, 10:43 Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution. If k is a positive integer and n = k(k + 7), is n divisible by 6? (1) k is odd. (2) When k is divided by 3, the remainder is 2. There are 2 variables (n,k) and 1 equation (n=k(k+7)) in the original condition, 2 more equations in the given conditions, so there is high chance (D) will be our answer. For condition 1, the answer is 'no' for k=1, but 'yes' for k=5, so this is insufficient. For condition 2, k=3m+2 (m is a positive integer), or k+7=3m+2+7=3(m+3)=a multiple of 3 and k(k+7). This is always even and a multiple of 3, which means its always divisible by 6. This answers the question 'yes'; this is sufficient, and the answer is B. For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously 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$79 for 1 month Online Course"
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Re: If k is a positive integer and n = k(k + 7k), is n divisible  [#permalink]

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27 Mar 2016, 09:06
VeritasPrepKarishma

Hello Karishma! A question here.. how do you quickly pull the 3 out of the initial statement?
(3b + 2)(3b + 9) = this one-> 3*(3b + 2)(b + 3)

Thank you!
Re: If k is a positive integer and n = k(k + 7k), is n divisible   [#permalink] 27 Mar 2016, 09:06

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