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# If x and y are integers, is x+y an even number?

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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GMAT 1: 760 Q51 V42
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If x and y are integers, is x+y an even number?  [#permalink]

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20 Apr 2018, 00:08
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45% (medium)

Question Stats:

61% (01:26) correct 39% (01:44) wrong based on 86 sessions

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[GMAT math practice question]

If $$x$$ and $$y$$ are integers, is $$x+y$$ an even number?

1) $$x+3y$$ is even
2) $$(x-1)(y-1)$$ is odd

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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Retired Moderator Joined: 27 Oct 2017 Posts: 1585 Location: India Concentration: International Business, General Management GPA: 3.64 WE: Business Development (Energy and Utilities) Re: If x and y are integers, is x+y an even number? [#permalink] ### Show Tags 20 Apr 2018, 00:42 1 Statement1: x+3y is even. Now as MULTIPLICATION BY AN ODD INTEGER DOESN’T CHANGE THE ODD/EVEN NATURE OF AN INTEGER. Hence x +y is even (same as x+3y or 3x+5y etc) SUFFICIENT Statement 2: (x-1)(y-1) is odd Since the product of two integers are odd only if both of them are odd, (x-1) is odd, hence x =odd+1 = even. Similarly, (y-1) is odd, hence y= odd+1 = even. Hence, x+y = even+even = even SUFFICIENT. Answer D _________________ GMAT Club Legend Joined: 11 Sep 2015 Posts: 4604 Location: Canada GMAT 1: 770 Q49 V46 Re: If x and y are integers, is x+y an even number? [#permalink] ### Show Tags 20 Apr 2018, 06:00 Top Contributor MathRevolution wrote: [GMAT math practice question] If $$x$$ and $$y$$ are integers, is $$x+y$$ an even number? 1) $$x+3y$$ is even 2) $$(x-1)(y-1)$$ is odd Target question: Is x+y an even integer? Statement 1: x+3y is even There are 4 possible cases to consider. Let's test them all Case a: x is ODD and y is ODD. Notice that x +3y = ODD + (3)(ODD) = ODD + ODD = EVEN. In this case, x + y = ODD + ODD = EVEN. So, the answer to the target question is YES, x+y IS even Case b: x is ODD and y is EVEN. Notice that x +3y = ODD + (3)(EVEN) = ODD + EVEN= ODD. This does NOT meet the requirement that x+3y is even, so IGNORE case b Case c: x is EVEN and y is ODD. Notice that x +3y = EVEN + (3)(ODD) = EVEN + ODD= ODD. This does NOT meet the requirement that x+3y is even, so IGNORE case c Case d: x is EVEN and y is EVEN. Notice that x +3y = EVEN + (3)(EVEN) = EVEN+ EVEN= EVEN. In this case, x + y = EVEN + EVEN = EVEN. So, the answer to the target question is YES, x+y IS even So, cases a and d are the only to possible cases. Since both cases yields the SAME answer to the target question (YES, x+y IS even), statement 1 is SUFFICIENT Statement 2: (x-1)(y-1) is odd If the product of two integers is ODD, then the two integers must each be ODD So, (x-1) must be ODD, and (y-1) must be ODD If (x-1) is ODD, then we can be certain that x is EVEN If (y-1) is ODD, then we can be certain that y is EVEN If x and y are both EVEN, then x+y = EVEN + EVEN = EVEN So, the answer to the target question is YES, x+y IS even Since we can answer the target question with certainty, statement 2 is SUFFICIENT Answer: D Cheers, Brent _________________ Test confidently with gmatprepnow.com Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8787 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If x and y are integers, is x+y an even number? [#permalink] ### Show Tags 22 Apr 2018, 17:27 => 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 (x and y) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first. Conditions 1) & 2) Since both conditions are satisfied only when x is even and y is even, x + y must be an even number. Thus, both conditions together are sufficient. Since this question is an integer 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) There are two ways in which x + 3y can be even. If x and y are both even, then x + y is even. If x and y are both odd, then x + y is even too. Thus, condition 1) is sufficient. Condition 2) Since (x-1)(y-1) is odd, both x -1 and y – 1 must be odd. So, x and y must both be even. It follows that x + y is even. Thus, condition 2) is sufficient too. Therefore, D is the answer. Answer: D Normally, in problems which require 2 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. _________________ 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 x and y are integers, is x+y an even number?   [#permalink] 22 Apr 2018, 17:27
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