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• ### $450 Tuition Credit & Official CAT Packs FREE December 15, 2018 December 15, 2018 10:00 PM PST 11:00 PM PST Get the complete Official GMAT Exam Pack collection worth$100 with the 3 Month Pack ($299) • ### FREE Quant Workshop by e-GMAT! December 16, 2018 December 16, 2018 07:00 AM PST 09:00 AM PST Get personalized insights on how to achieve your Target Quant Score. # If M and N are integers, is ((10^M)) + N)/3 an integer?  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: ### Hide Tags Manager Joined: 17 Jul 2009 Posts: 239 Concentration: Nonprofit, Strategy GPA: 3.42 WE: Engineering (Computer Hardware) If M and N are integers, is (10^M + N)/3 an integer? [#permalink] ### Show Tags Updated on: 27 Jul 2015, 13:45 1 10 00:00 Difficulty: 85% (hard) Question Stats: 43% (01:47) correct 57% (01:36) wrong based on 281 sessions ### HideShow timer Statistics If M and N are integers, is (10^M + N)/3 an integer? (1) N = 5 (2) MN is even Originally posted by sprtng on 06 Aug 2009, 19:56. Last edited by Bunuel on 27 Jul 2015, 13:45, edited 1 time in total. Renamed the topic, edited the question and added the OA. ##### Most Helpful Expert Reply Math Expert Joined: 02 Sep 2009 Posts: 51218 Re: If M and N are integers, is ((10^M)) + N)/3 an integer? [#permalink] ### Show Tags 24 Nov 2010, 01:44 4 1 chiragatara wrote: If M and N are integers, is ((10^M)) + N) / 3 an integer? 1. N = 5 2. MN is even Answer to question is E. From statement 1 and 2, M has to be Even Positive as N = 5 and MN is even. Now when 5 is added to 10^M where M is even positive and then divided by 3 it always results into an integer. Basically the the question ask whether $$10^m+n$$ is divisible by 3. Now, in order $$10^m+n$$ to be divisible by 3: A. It must be an integer, and B. the sum of its digits must be multiple of 3. (1) N = 5 --> if $$m<0$$ (-1, -2, ...) then $$10^m+n$$ won't be an integer at all (for example if $$m=-1$$ --> $$10^m+n=\frac{1}{10}+5=\frac{51}{10}\neq{integer}$$), thus won't be divisible by 3, but if $$m\geq{0}$$ (0, 1, 2, ...) then $$10^m+n$$ will be an integer and also the sum of its digits will be divisible by 3 (for example for $$m=1$$ --> $$10^m+n=10+5=15$$ --> 15 is divisible by 3). Not sufficient. (2) MN is even --> clearly insufficient, as again $$m$$ can be -2 and $$n$$ any integer and the answer to the question will be NO or $$m$$ can be 0 and $$n$$ can be 2 and the answer to the question will be YES. Not sufficient. (1)+(2) From $$mn=even$$ and $$n=5$$ it's still possible for $$m$$ to be negative even integer (-2, -4, ...), so $$10^m+n$$ may or may not be divisible by 3. Not sufficient. Answer: E. _________________ ##### General Discussion Manager Joined: 18 Jul 2009 Posts: 164 Location: India Schools: South Asian B-schools Re: If M and N are integers, is (10^M + N)/3 an integer? [#permalink] ### Show Tags 06 Aug 2009, 21:59 1 A Statement 1: if u take N=5 and any value for M it will always hold true that is (10^M + N)/3 an integer; becoz 10^M ( let M = any value) will always leave a reminder of 1....and 1+5 = 6 which is divisable by 3 ....sufficient Statement 2 : if MN is even ...this means (M,N) can be both even or odd (except both being odd at a time)....if taken an example.... (M,N) => (1,2) => 10+2 =12/3 => integer (M,N) => (1,4) => 10+4 =14/3 => non-integer hence insufficient if u like my post....consider it for Kudos _________________ Bhushan S. If you like my post....Consider it for Kudos CEO Joined: 17 Nov 2007 Posts: 3440 Concentration: Entrepreneurship, Other Schools: Chicago (Booth) - Class of 2011 GMAT 1: 750 Q50 V40 Re: If M and N are integers, is (10^M + N)/3 an integer? [#permalink] ### Show Tags 06 Aug 2009, 22:13 1 1 Let's consider $$10^M$$ at M>=0: $$10^M = {1, 10, 100, 1000....}$$ or $$3k+1$$ at M<0: 10^M is a fraction and our expression is not an integer. Now, let's see our statements: 1) N=5 at M>=0: (3k+1 + 5) /3 = k+2 - an integer at M<0: a fraction. insufficient 2) MN is even a) at M=2, N=5 our expression is an integer b) at M=-2, N=-5 our expression in a fraction insufficient 1)&2) N=5 & MN is even --> N=5, M is even For all even M>=0 we will get an integer. Now we have interesting question: Could negative numbers be even or odd? if yes, we can choose M=-2, N=5 and get a fraction. if no, M cannot be negative and two statements are sufficient. Judging by http://en.wikipedia.org/wiki/Even_and_odd_numbers negative integers can be classified as odd/even. So, our answer is insufficient. E _________________ HOT! GMAT TOOLKIT 2 (iOS) / GMAT TOOLKIT (Android) - The OFFICIAL GMAT CLUB PREP APP, a must-have app especially if you aim at 700+ | Limited GMAT/GRE Math tutoring in Chicago Intern Joined: 07 Dec 2009 Posts: 12 Re: If M and N are integers, is (10^M + N)/3 an integer? [#permalink] ### Show Tags 17 Dec 2009, 19:25 how did you get 3k + 1? can someone please explain? thanks. Manager Joined: 12 Oct 2008 Posts: 50 Re: If M and N are integers, is (10^M + N)/3 an integer? [#permalink] ### Show Tags 18 Dec 2009, 20:18 10^K = 3k+1: for any K>=0 It means any number which is 10^K i.e. 1,10,100,1000 ..., when divide by 3, gives remainder 1. Manager Joined: 26 Nov 2009 Posts: 122 Re: If M and N are integers, is (10^M + N)/3 an integer? [#permalink] ### Show Tags 23 Jan 2010, 09:56 Thanks I committed 2 careless mistakes Initially I chose A ignoring the fact M and are just integers not positive integers and then I chose C since MN is even I just assumed it is positive. but the correct ans is E since both statements would be insufficient to ans the question Intern Joined: 12 Aug 2010 Posts: 13 If M and N are integers, is ((10^M)) + N)/3 an integer? [#permalink] ### Show Tags 24 Nov 2010, 00:44 1 If M and N are integers, is ((10^M)) + N)/3 an integer? (1) N = 5 (2) MN is even Answer to question is . From statement 1 and 2, M has to be Even Positive as N = 5 and MN is even. Now when 5 is added to 10^M where M is even positive and then divided by 3 it always results into an integer. M25-07 Manager Joined: 26 Apr 2010 Posts: 108 Concentration: Strategy, Entrepreneurship Schools: Fuqua '14 (M) Re: If M and N are integers, is ((10^M)) + N)/3 an integer? [#permalink] ### Show Tags 24 Nov 2010, 07:53 Thanks for the question...I missed the negative case in this question as well. Bunuel, thanks for solving. _________________ I appreciate the kudos if you find this post helpful! +1 Intern Joined: 28 Jun 2013 Posts: 3 Re: If M and N are integers, is ((10^M)) + N)/3 an integer? [#permalink] ### Show Tags 19 Sep 2013, 21:22 Hi, "I read in a forum that negative numbers cannot be tagged as odd or even. So, if I get a DS question stating that X is a even integer, then it also implies that X>=0." However, in the question given below, the logic is not holding true. I was considering the right answer to be C), because if MN = even and N =5 , then it implies that M must be odd integer (and hence M is > 0). Can you clarify if the argument above is false? If M and N are integers, is 10M+N3 an integer? (1) N=5 (2) MN is even Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.EACH statement ALONE is sufficient.Statements (1) and (2) TOGETHER are NOT sufficient.Mark as a guessHide Answer The question does not mention whether M and N are positive. Hence, the statements taken together are not sufficient because the answer is YES if M=2, N=5 and NO if M=−2; N=5. The correct answer is E Senior Manager Joined: 24 Aug 2009 Posts: 469 Schools: Harvard, Columbia, Stern, Booth, LSB, Re: If M and N are integers, is ((10^M)) + N)/3 an integer? [#permalink] ### Show Tags 19 Sep 2013, 22:44 SurabhiStar wrote: Hi, "I read in a forum that negative numbers cannot be tagged as odd or even. So, if I get a DS question stating that X is a even integer, then it also implies that X>=0." However, in the question given below, the logic is not holding true. I was considering the right answer to be C), because if MN = even and N =5 , then it implies that M must be odd integer (and hence M is > 0). Can you clarify if the argument above is false? If M and N are integers, is 10M+N3 an integer? (1) N=5 (2) MN is even First, every number except zero "0" can be categorized either Even or odd. Secondly, "X is a even integer" does not necessarily mean X>=0. X can be smaller than 0 as well. _________________ If you like my Question/Explanation or the contribution, Kindly appreciate by pressing KUDOS. Kudos always maximizes GMATCLUB worth -Game Theory If you have any question regarding my post, kindly pm me or else I won't be able to reply Math Expert Joined: 02 Sep 2009 Posts: 51218 Re: If M and N are integers, is ((10^M)) + N)/3 an integer? [#permalink] ### Show Tags 19 Sep 2013, 23:57 fameatop wrote: SurabhiStar wrote: Hi, "I read in a forum that negative numbers cannot be tagged as odd or even. So, if I get a DS question stating that X is a even integer, then it also implies that X>=0." However, in the question given below, the logic is not holding true. I was considering the right answer to be C), because if MN = even and N =5 , then it implies that M must be odd integer (and hence M is > 0). Can you clarify if the argument above is false? If M and N are integers, is 10M+N3 an integer? (1) N=5 (2) MN is even First, every number except zero "0" can be categorized either Even or odd. Secondly, "X is a even integer" does not necessarily mean X>=0. X can be smaller than 0 as well. 1. EVEN/ODD An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder. An odd number is an integer that is not evenly divisible by 2. According to the above both negative and positive integers can be even or odd. 2. ZERO Zero is an even integer. Zero is nether positive nor negative, but zero is definitely an even number. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder and as zero is evenly divisible by 2 then it must be even (in fact zero is divisible by every integer except zero itself). Hope it helps. _________________ EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 13087 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: If M and N are integers, is ((10^M)) + N)/3 an integer? [#permalink] ### Show Tags 18 Jan 2015, 12:06 Hi All, This DS question can be solved by TESTing VALUES, but you have to thorough with your TESTs and thinking. We're told that M and N are INTEGERS. We're asked if (10^M + N)/3 is an integer. This is a YES/NO question. Fact 1: N = 5 IF.... M = 0 N = 5 6/3 IS an integer and the answer to the question is YES. IF... M = 1 N = 5 15/3 IS an integer and the answer to the question is YES. At this point, it might be tempting to say that Fact 1 is sufficient, but we have NOT yet considered ALL TYPES of integers.... IF... M = -1 N = 5 5.1/3 is NOT an integer and the answer to the question is NO. Fact 1 is INSUFFICIENT Fact 2: MN is EVEN IF... M = 0 N = 5 6/3 is an integer and the answer to the question is YES. IF.... M = 0 N = 1 2/3 is NOT an integer and the answer to the question is NO. Fact 2 is INSUFFICIENT Combined, we know.... N = 5, MN is EVEN IF.... M = 0 N = 5 6/3 is an integer and the answer to the question is YES. IF.... M = -2 N = 5 5.01/3 is NOT an integer and the answer to the question is NO. Combined, INSUFFICIENT Final Answer: GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com # Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
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Re: If M and N are integers, is ((10^M)) + N)/3 an integer?  [#permalink]

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18 Jan 2015, 23:42
Nice question....

Initially didnt observe that M can be negative....then it looked too easy....well had it come in the early part of real test....i wud have thought im still in easy qtns zone....and put it as A

Thanks for posting the question
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Re: If M and N are integers, is ((10^M)) + N)/3 an integer?  [#permalink]

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19 Jan 2015, 00:03
1
chiragatara wrote:
If M and N are integers, is ((10^M)) + N)/3 an integer?

(1) N = 5
(2) MN is even

From statement 1 and 2, M has to be Even Positive as N = 5 and MN is even.

Now when 5 is added to 10^M where M is even positive and then divided by 3 it always results into an integer.

M25-07

Question: Is ((10^M)) + N)/3 an integer?
Re-worded: Is (10^M + N) a multiple of 3?

Note that 10^M will be something like 1000... if M is non-negative. Then, 10^M will be of the form (3a+1) because it will leave remainder 1 when divided by 3. If M is negative, 10^M will not be an integer.

Assuming M is non-negative, for 10^M + N to be a multiple of 3, N should be of them form 3n+2.

(1) N = 5
We don't know whether M is non-negative. Not sufficient.

(2) MN is even
again, we don't know whether M is non-negative. Not sufficient.

Using both, we know that M is even since N = 5 (odd) but we don't know whether it is non-negative.

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Re: If M and N are integers, is ((10^M)) + N)/3 an integer?  [#permalink]

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28 Aug 2017, 02:57
VeritasPrepKarishma wrote:
chiragatara wrote:
If M and N are integers, is ((10^M)) + N)/3 an integer?

(1) N = 5
(2) MN is even

From statement 1 and 2, M has to be Even Positive as N = 5 and MN is even.

Now when 5 is added to 10^M where M is even positive and then divided by 3 it always results into an integer.

M25-07

Question: Is ((10^M)) + N)/3 an integer?
Re-worded: Is (10^M + N) a multiple of 3?

Note that 10^M will be something like 1000... if M is non-negative. Then, 10^M will be of the form (3a+1) because it will leave remainder 1 when divided by 3. If M is negative, 10^M will not be an integer.

Assuming M is non-negative, for 10^M + N to be a multiple of 3, N should be of them form 3n+2.

(1) N = 5
We don't know whether M is non-negative. Not sufficient.

(2) MN is even
again, we don't know whether M is non-negative. Not sufficient.

Using both, we know that M is even since N = 5 (odd) but we don't know whether it is non-negative.

should the answer not be A as statement one tells us that the expression will be an integer. the question asks if the expression will be an integer and it is clearly visible that the statement 1 with N=5 will not leave a remainder and hence be divisible and hence will be an integer. Is this not sufficient to answer. the expression can also be written as (10^m)/3 + (N/3) = (1^m)/3 + 5/3= remainder of 0 and hence will be an integer. theoretically this should suffice, however, plugging values is a different result. please clear the doubt.
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Re: If M and N are integers, is ((10^M)) + N)/3 an integer?  [#permalink]

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28 Aug 2017, 15:28
Hi kaulshanx,

The prompt tells us that M and N are integers. It does NOT state that they are necessarily positive integers though. If you assume that M and N are positive, then your deduction is correct. However, what if M = -1.....? That IS an integer, but it will NOT lead to an integer result when you answer the question.

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Re: If M and N are integers, is ((10^M)) + N)/3 an integer?  [#permalink]

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11 Aug 2018, 13:37
I made one of the classic mistake of considering M only as a positive integer when i read M and N are integers and chose option A incorrectly.

Good question
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Re: If M and N are integers, is ((10^M)) + N)/3 an integer?  [#permalink]

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13 Aug 2018, 11:58
statement 1:
10^1+5/3=Integer
10^-1+5=(1/10+5)/3=Not an integer
INSUFFICENT

Statement 2:
Clearly insufficent

1&2, clearly insufficent

Hence---> E.
Re: If M and N are integers, is ((10^M)) + N)/3 an integer? &nbs [#permalink] 13 Aug 2018, 11:58
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