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• ### $450 Tuition Credit & Official CAT Packs FREE November 15, 2018 November 15, 2018 10:00 PM MST 11:00 PM MST EMPOWERgmat is giving away the complete Official GMAT Exam Pack collection worth$100 with the 3 Month Pack ($299) # If i and j are integers, is i + j an even integer?  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: ### Hide Tags Math Expert Joined: 02 Sep 2009 Posts: 50572 If i and j are integers, is i + j an even integer? [#permalink] ### Show Tags 06 Aug 2012, 02:13 1 6 00:00 Difficulty: 5% (low) Question Stats: 87% (00:38) correct 13% (00:42) wrong based on 1132 sessions ### HideShow timer Statistics If i and j are integers, is i + j an even integer? (1) i < 10 (2) i = j Practice Questions Question: 14 Page: 276 Difficulty: 550 _________________ Math Expert Joined: 02 Sep 2009 Posts: 50572 Re: If i and j are integers, is i + j an even integer? [#permalink] ### Show Tags 06 Aug 2012, 02:13 1 SOLUTION If i and j are integers, is i + j an even integer? (1) i < 10. Clearly insufficient. (2) i = j --> $$i + j=i+i=2i=2*integer=even$$. Sufficient. Answer: B. _________________ Manager Joined: 05 Jul 2012 Posts: 69 Location: India Concentration: Finance, Strategy GMAT Date: 09-30-2012 GPA: 3.08 WE: Engineering (Energy and Utilities) Re: If i and j are integers, is i + j an even integer? [#permalink] ### Show Tags 06 Aug 2012, 07:08 1 In Easiest form 1) i<10 7 + 12 & 7+ 13 even & odd 2) i = j both either 2k or 2k+1 .. 2k + 2k = eveen ..... 2k+1 + 2k+1 = 4k+2 = even. Intern Joined: 26 Nov 2014 Posts: 2 Re: If i and j are integers, is i + j an even integer? [#permalink] ### Show Tags 17 Dec 2014, 05:15 What if both i and j are negative ? For example -3+(-3)= -6. Can we categorise negative numbers into even or odd ? Shouldn't the answer be E? Math Expert Joined: 02 Sep 2009 Posts: 50572 Re: If i and j are integers, is i + j an even integer? [#permalink] ### Show Tags 17 Dec 2014, 05:20 1 aseshank wrote: What if both i and j are negative ? For example -3+(-3)= -6. Can we categorise negative numbers into even or odd ? Shouldn't the answer be E? Yes. 1. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder. Since -6 IS divisible by 2 (-6/2 = -3), then -6 is an even integer. Even integers are: ..., -6, -4, -2, 0, 2, 4, 6, 8, ... 2. An odd number is an integer that is not evenly divisible by 2. Theory on Number Properties: math-number-theory-88376.html Tips on Number Properties: number-properties-tips-and-hints-174996.html All DS Number Properties Problems to practice: search.php?search_id=tag&tag_id=38 All PS Number Properties Problems to practice: search.php?search_id=tag&tag_id=59 _________________ Target Test Prep Representative Status: Head GMAT Instructor Affiliations: Target Test Prep Joined: 04 Mar 2011 Posts: 2830 If i and j are integers, is i + j an even integer? [#permalink] ### Show Tags 30 Jul 2016, 07:34 1 sirrock wrote: This topic is locked. If you want to discuss this question please re-post it in the respective forum. If i and j are integers, is i + j an even integer? (1) i < 10 (2) i = j We need to determine whether i + j is an even integer. Remember: even + even = even odd + odd = even Thus, if we can determine that i and j are either both even or both odd, we will be able to answer the question. Statement One Alone: i < 10 Knowing that i is less than 10 is not enough to determine whether i + j is an even integer. Statement one alone is not sufficient to answer the question. We can eliminate answer choices A and D. Statement Two Alone: i = j Since we know i = j, we know that i and j are both even or both odd. Following our addition rules for even and odd numbers, we see that i + j must be even. Statement two alone is sufficient to answer the question. The answer is B. _________________ Jeffery Miller Head of GMAT Instruction GMAT Quant Self-Study Course 500+ lessons 3000+ practice problems 800+ HD solutions Current Student Joined: 12 Aug 2015 Posts: 2633 Schools: Boston U '20 (M) GRE 1: Q169 V154 Re: If i and j are integers, is i + j an even integer? [#permalink] ### Show Tags 03 Dec 2016, 17:16 Here is my solution -> Notice i and j are integers (this info is very important in even/odd Questions) We need to check the even/odd nature of i+j Notice i+j will be even when either both i and j are even or both i and j are odd Lets dive into statements Statement 1 i<10 no clue of the even /odd nature of i and j Additionally we must remember that negative integers can be even/odd too. Hence Insufficient Statement 2 i=j so i+j=2i,which is always even Hence sufficient Hence B _________________ MBA Financing:- INDIAN PUBLIC BANKS vs PRODIGY FINANCE! Getting into HOLLYWOOD with an MBA! The MOST AFFORDABLE MBA programs! STONECOLD's BRUTAL Mock Tests for GMAT-Quant(700+) AVERAGE GRE Scores At The Top Business Schools! Manager Status: On a 600-long battle Joined: 21 Apr 2016 Posts: 138 Location: Hungary Concentration: Accounting, Leadership Schools: Erasmus '19 GMAT 1: 410 Q18 V27 GMAT 2: 490 Q35 V23 Re: If i and j are integers, is i + j an even integer? [#permalink] ### Show Tags 02 Apr 2017, 21:46 Bunel's explanation is very handy in case you don't recall the odd/even rules. If you recall that Odd + Odd = Even and Even + Even = Even, statements two allows you to answer the question confidently without any calculation. _________________ "When the going gets tough, the tough gets going!" |Welcoming tips/suggestions/advices (you name it) to help me achieve a 600| EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 12852 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: If i and j are integers, is i + j an even integer? [#permalink] ### Show Tags 12 Dec 2017, 11:10 Hi All, We're told that I and J are integers. We're asked if(I + J) is an even integer. This is a YES/NO question. We can solve it by TESTing VALUES. 1) I < 10 IF... I = 9 and J = 1, then the answer to the question is YES. I = 9 and J = 2, then the answer to the question is NO. Fact 1 is INSUFFICIENT 2) I = J Fact 2 tells us that I and J are the SAME integer, so (I+J) can be rewritten as either 2I or 2J. Since the product of any integer and 2 results in an EVEN integer, the answer to the question is ALWAYS YES. You can also prove it with examples: I = 1 and J = 1 ... total = 2 I = 2 and J = 2 ... total = 4 I = 3 and J = 3 ... total = 6 Etc. Fact 2 is SUFFICIENT 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 i and j are integers, is i + j an even integer?  [#permalink]

### Show Tags

14 Dec 2017, 20:08
Bunuel wrote:
If i and j are integers, is i + j an even integer?

(1) i < 10
(2) i = j

Practice Questions
Question: 14
Page: 276
Difficulty: 550

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 2 variables (i and j) and 0 equations, C is most likely to be the answer and so we should consider 1) & 2) first.

Conditions 1) & 2)

i + j = i + i = 2i.
It is an even integer.

Since this question is a statistics question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
i = 1, j = 9 : i + j = 10 is an even integer.
i = 1, j = 8 : i + j = 9 is not an even integer.
The condition 1) is not sufficient.

Condition 2)
i + j = i + i = 2i.
It is an even integer.
The condition 2) only is sufficient.

Normally, in problems which require 2 or more additional equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
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Re: If i and j are integers, is i + j an even integer? &nbs [#permalink] 14 Dec 2017, 20:08
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