Even-Odd numbers is deemed to be among the easier concepts on the GMAT Quant, and yet, come 700+ level questions from this concept, many students get them wrong. We have seen closely studied the mistakes that students make in Even-Odd questions – from the doubts they ask in Even-Odd questions in our internal forums, from the mistakes made by 1000+ students in our recurring Number Properties Live Classroom, and most recently, in *The E-GMAT Number Properties Knockout*** **that was attempted more than 5000 times.

- Getting intimidated by complex expressions
- Wasting time on unimportant terms
- Getting stumped in division

In this article, we will explain each pitfall with examples, discuss why it is important to avoid that pitfall, tell you how to avoid it and finally, give you 700+ level practice questions.

Sounds good? So read on, to make sure that you’ll never be among the unlucky students who err in Even-Odd questions. Amen to that!

A few Even-Odd questions may have scary-looking expressions. For example, consider this question

**P1.1**** If j is a positive integer, is (j ^{3}-27)^{2}(j^{3}+1)^{3} odd?**

Did you feel a bit nervous reading this question? Well, that is the first pitfall that you have to guard against. Because, if you let yourself become nervous, you will:

- Either leave the question without answering
- Or you will panic; panic clouds our ability to think rationally and so, increases our chances of making an error.

- For example, in your panic, you may scramble to remember and apply the formula for a
^{3}+ b^{3}on the terms of this expression, and then, realize, much to your dismay, that you’ve complicated the question even further L

So, as you can see, ‘getting intimidated by complex expressions’ is indeed a dangerous pitfall.

The next time you face such a question and notice your heartbeat increasing, take a deep breath and tell yourself,

*“Since this is a GMAT question, it can be simplified elegantly.”*

This is true! The beauty of official GMAT questions is that no matter how complex they look, they can always be simplified to a couple of cases.

*So, let’s think through the question we posed above and see how it can be simplified.*

*1 ^{st} Simplification *

*The given expression is (j ^{3}-27)^{2}(j^{3}+1)^{3}*

*You’re probably familiar with the property that **the power of a number doesn’t impact the even-odd nature of the number.*

*(Even)*^{n}, where n is a positive integer = Even*Similarly, (Odd)*^{n}= Odd

*So, *

*(j*^{3}– 27)^{2}will have the same even-odd nature as (j^{3}– 27). Similarly, (j^{3}+ 1)^{3}will have the same even-odd nature as (j^{3}+1)*j*^{3}will have the same even-odd nature as j itself.

*So, using this property, we’ve done the first level of simplification: now, we only have to determine the even-odd nature of this, simpler expression: (j-27)(j + 1)*

* *

*2 ^{nd} Simplification *

*The simpler expression is a product of 2 terms: (j – 27) and (j+1)*

*When will the product of 2 terms be odd? Only if each of the 2 terms are themselves odd. If even one of these terms is even, the product will be even.*

*So, to answer the question, we need to know: are each of the 2 terms odd?*

*So, from the earlier situation of dealing with the product as a whole, we are now dealing with individual terms only: (j – 27) and (j + 1)*

*Getting to the answer*

*Now, j can either be Even or Odd.*

* *

*Case 1: j is odd*

*In this case, j + 1 = Odd + Odd = Even*

*And j - 27 = Odd - Odd = Even*

*Since both the terms are Even, the answer in this case will be NO, the given expression in not odd.*

*Case 2: j is even*

*In this case, j + 1 = Even + Odd = Odd*

*And, j - 27 = Even - Odd = Odd*

*Since both the terms are odd, the answer in this case will be YES, the given expression is odd*

*So, as you can see, using this step-wise approach, we’ve been able to simplify the question to this:*

* Is j even?*

**Use the properties of Even-Odd combinations to simplify scary-looking expressions.** Have the confidence that all Even-Odd questions in the GMAT can be easily simplified. Don’t get intimidated by complex expressions in Even-Odd questions and avoid the impulse to search for algebraic formulae to apply on such expressions.

You’ll know that you’ve learnt this lesson well, if your heart doesn’t skip a beat at the first look of the following question:

**P1.2**** If X = P*N ^{K} + P where N and K are positive integers, is X divisible by 2?**

**(1) ****N + KN = 915**

**(2) ****P ^{35} + 35^{P} is Even**

(The detailed solution of this question is available here)

What we mean by ‘unimportant terms’ is ‘the terms that do not impact the Even- Odd nature of the expression. For example, consider the following question:

**P2.1**** If a and b are integers, is a + 8b even?**

In this expression, the term 8b will be even, irrespective of whether b is even or odd (because, Even*Odd = Even and Even*Even = Even). So, you should focus all your attention on analysing whether a is even or odd, because that is what will get you to the answer.

If you fall into the pitfall of analysing the given information to determine the even-odd nature of ** b**, then you’ll be squandering your most precious resource in the GMAT – Time. Minutes frittered away thus may create a time crunch towards the end of the test, and then, coming under the pressure of the seconds ticking away, you may frantically answer even questions that you know, wrong. So, it is very important to be on strict guard against even a moment spent on unneeded analysis. And, in Even-Odd questions, it’s all too easy to fall into this booby trap.

In order to not waste even a second on the unimportant terms, here are a few pointers that you should use to weed out the unimportant terms in an expression:

- A term of the form (Even number)*(X) will always be even
- In a term of the form (Even number) + X, the (Even number) plays no role in the Even-Odd nature of the term
- In a term of the form (Odd number)*(X), the (Odd number) plays no role in the Even-Odd nature of the term

*You’ve already seen an example of the first pointer in Question P2.1*

*Here’s an example that will show all the three pointers in action*

**P2.2**** If a, b, c and n are integers, is a + 8b + (2n+1)c even?**

*1 ^{st} Pointer *

*The term 8b will always be even, irrespective of the value of b*

*2 ^{nd} Pointer *

*In the given expression, the even term 8b doesn’t impact the even-odd nature of this expression. So, the expression will have the same even-odd nature as the sum a + (2n+1)c*

*3 ^{rd} Pointer*

*In the term (2n+1)c, (2n+1) is an odd number, and so plays no role in the even-odd nature of this term. So, the term (2n+1)c will have the same even-odd nature as c.*

*So, the expression a + (2n+1)c will have the same even-odd nature as the expression a + c*

To some students Pitfall 2 may seem similar to Pitfall 1 because the strategy suggested to avoid Pitfall 2 (the Three Pointers) also leads to simplification of the given expression. However, even though the *effect *of the strategies suggested in Pitfalls 1 and 2 may be the same, the *problems *that these strategies tackle are different. In Pitfall 1, the problem is that a student may get intimidated by a difficult-looking expression. In Pitfall 2, on the other hand, the problem is that a student may waste time on analysing terms that do not contribute to the Even-Odd nature of an expression. These are two distinct problems, and so, Pitfalls 1 and 2 are distinct as well.

**When you see an expression, first use the Three Pointers to determine the unimportant terms. **Do not waste precious time on processing the unimportant terms.

See how much time you take on this question and if you waste time on any term that doesn’t deserve it:

**P2.3**** ****If a, b and n are positive integers such that n = 3a – b ^{3}, is n^{2} + 3 divisible by 2?**

**(1) ****a ^{2} – 4b^{3} – 5 = 0**

**(2) 3b ^{3} – a^{2} + 6 = 0 **

(The detailed discussion of this question is available here)

If A and B are given to be integers, where A > B and A/B is an integer, can you smoothly work out the relation between the even-odd nature of A, B and the integer A/B?

For example, consider the following question:

**P3.1**** If A, B and X are integers, X/B is an even integer and XB/(4A+1) is an integer, is XB/(4A+1) odd?**

If you don’t have a firm approach to deal with this and similar questions, you’re bound to feel flummoxed, and then you’ll:

- Either give up this question as too difficult
- Or will gingerly try number substitution to see which values of X and B give an even value of X/B, and then with those values of X, try to see if XB/(4A+1)

Both possible actions are costly – in terms of lost score points and lost time. So, it is important to not fall prey to such questions.

This pitfall is easily avoided by following the standard approach presented here –

**Convert the division equation into a multiplication equation.**

*Let’s illustrate this approach on question P3.1*

*The division equation that we can write for the terms X/B is:*

*We can convert this equation into a multiplication equation by multiplying both sides with B. We get:*

*X = (Even number)*B*

--> *X is Even (Refer to Pointer (i) in Pitfall 2)*

*Now, let’s write the division equation for the term (XB/4A+1):*

XB/(4A + 1) = integer Z (say)

*Converting this equation into a multiplication equation, we get:*

--> *XB = (4A+1)*(Z)*

--> *XB has the same Even-Odd nature as Z **(because 4A + 1 is odd – Refer to Pointers (ii) and (iii) in Pitfall 2)*

*Since X is Even, XB is Even*

--> *Z is Even*

*So, we see that the given expression will be Even.*

In Even-Odd questions that involve division, convert the division equation into multiplication equation.

**P3.2**** If x, y and z are positive integers such that x^{4 }y^{3} = z^{2}, is x^{9} – y^{6} odd?**

**(1) (x^{4} y^{3})/(x^{2} + y^{2})can be written in the form 4k + 3, where k is a positive integer.**

**(2) z = x + y**

(The detailed discussion of this question is available here)

Even when you know a concept, you might not be able to answer the questions that test advanced application of that concept. In this article, we saw the three pitfalls that many students fall into in Even-Odd questions. If you make a conscious effort to avoid these pitfalls, you’ll find that your ability to answer 700+ level Even-Odd questions will improve significantly. As a happy co-benefit, the time you take to solve the questions will also come down.

If you wish to work further on the 3 pitfalls, please practice the 3 questions provided below. If you feel you need more help with this concept, please go to our Free Trial.

Wish you enjoy your journey of GMAT Prep and reach a great score on the GMAT!

**Is the product of two integers A and B odd?**

**(1) ****A is the number of factors of N, where N is a perfect square and B = A ^{3} -1**

**(2) ****A is a product of two consecutive prime numbers and when **** is added to A, the sum is an odd number.**

(The detailed solution of this question is available here)

**If P and Q are positive integers, is the product 3P ^{Q} divisible by 2?**

**(1) 6Q ^{3} + 2 is an even number**

**(2) P + 8Q ^{2} is a prime number**

(The detailed solution of this question is available here )

**Is 3a + 2b + 5c even if 0<a<b<c and a, b and c are integers?**

**(1) 9a+7c is not even**

**(2) a ^{3}*(c-1)^{2} is odd**

(The detailed solution of this question is available here)

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This article was first published on the official blog of EXPARTUS, the first MBA admissions consulting firm to use personal branding as a key part of the b-school application process.

If , then the sum of all possible solutions for is:

A.

B.

C.

D.

E.

Question Discussion & Explanation

**Correct Answer** - D - (click and drag your mouse to see the answer)

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**Verbal (CR)**

Researchers in Germany have unearthed 400,000-year-old wooden spears from what [u]it appears was an ancient lakeshore hunting ground as stunning evidence of human ancestors who[/u] systematically hunted big game much earlier than believed.

a) it appears was an ancient lakeshore hunting ground as stunning evidence of human ancestors who

b) it appears had been an ancient lakeshore hunting ground and stunning evidence that human ancestors

c) appears was an ancient lakeshore hunting ground and is stunning evidence that human ancestors

d) appears to be an ancient lakeshore hunting ground, stunning evidence that human ancestors

e) appears that it is an ancient lakeshore hunting ground, stunning evidence of human ancestors who

Question Discussion & Explanation

**Correct Answer** - D - (click and drag your mouse to see the answer)

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Between 12:24 and 14:36 the hour hand of the clock turns by

A. 60 degrees

B. 66 degrees

C. 72 degrees

D. 74 degrees

E. 76 degrees

Question Discussion & Explanation

**Correct Answer** - B - (click and drag your mouse to see the answer)

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**Verbal (CR)**

In recent years cattle breeders have increasingly used crossbreading, [u]in part that their steers should acquire certain characteristics[/u] and partly because crossbreading is said to provide hybrid vigor.

A. in part that their steers should acquire certain characteristics

B. in part for the acquisition of certain characteristics in their steers

C. partly because of their steers acquiring certain characteristics

D. partly because certain characteristics should be acquired by their steers

E. partly to acquire certain characteristics in their steers

Question Discussion & Explanation

**Correct Answer** - E - (click and drag your mouse to see the answer)

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Set consists of a certain number of even integers divisible by 3. Is standard deviation of positive?

(1) All elements of set are positive.

(2) The range of set is 0.

Click and drag your mouse to see the answer.

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**Verbal (SC)**

Because of less availability and greater demand for scientific research, platinum remains consistently expensive, like gold.

(A) Because of less availability and greater demand for scientific research, platinum remains consistently expensive, like gold.

(B) Because of less availability and increased demand for scientific research, platinum remains consistently expensive, like that of gold.

(C) Because of decreased availability and increased demand in scientific research, platinum remains expensive, like gold.

(D) Because of decreased availability and increased demand for scientific research, platinum remains expensive, like gold.

(E) Because of decreased availability and greater demand in scientific research, platinum remains at a consistently high price, like that of gold.

Question Discussion & Explanation

**Correct Answer** - C - (click and drag your mouse to see the answer)

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A draining pipe can empty a pool in 4 hours. On a rainy day, when the pool is full, the draining pipe is opened and the pool is emptied in 6 hours. If rain inflow into the pool is 3 liters per hour, what is the capacity of the pool?

A. liters

B. liters

C. liters

D. liters

E. liters

Question Discussion & Explanation

**Correct Answer** - D - (click and drag your mouse to see the answer)

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**Verbal (SC)**

During the past decade, the labor market in France has not been operating according to free market principles, but instead stifling functioning through its various government regulations restricting the hiring and firing of workers.

(A) principles, but instead stifling functioning through its various government regulations restricting the hiring and firing of workers

(B) principles, instead it has been functioning in a stifled manner as a result of various government regulations that restrict the hiring and firing of workers

(C) principles, rather functioning despite being stifled as a result of government regulations that variously restrict worker hiring and firing

(D) principles; the hiring and firing of workers is restricted there by various government regulations, its functioning being stifled

(E) principles; instead, its functioning has been stifled by various government regulations restricting the hiring and firing of workers

Question Discussion & Explanation

**Correct Answer** - E - (click and drag your mouse to see the answer)

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A sphere is inscribed in a cube with an edge of centimeters. In terms of x what is the shortest possible distance from one of the vertices of the cube to the surface of the sphere?

A.

B.

C.

D.

E.

Question Discussion & Explanation (click and drag your mouse to see the answer)

**Correct Answer** - D -

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**Verbal (SC)**

Framed by traitorous colleagues, Alfred Dreyfus was imprisoned for twelve years before there was exoneration and his freedom.

(A) there was exoneration and his freedom

(B) he was to be exonerated with freedom

(C) being exonerated and freed

(D) exoneration and his freedom

(E) being freed, having been exonerated

Question Discussion & Explanation

**Correct Answer** - C - (click and drag your mouse to see the answer)

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If and are integers, is an integer?

(1)

(2) is even

Question Discussion & Explanation

**Correct Answer** - E - (click and drag your mouse to see the answer)

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**Verbal (CR)**

Exports of United States wood pulp will rise considerably during this year. The reason for the rise is that the falling value of the dollar will make it cheaper for paper manufacturers in Japan and Western Europe to buy American wood pulp than to get it from any other source.

Which of the following is an assumption made in drawing the conclusion above?

(A) Factory output of paper products in Japan and Western Europe will increase sharply during this year.

(B) The quality of the wood pulp produced in the United States would be adequate for the purposes of Japanese and Western European paper manufacturers.

(C) Paper manufacturers in Japan and Western Europe would prefer to use wood pulp produced in the United States if cost were not a factor.

(D) Demand for paper products made in Japan and Western Europe will not increase sharply during this year.

(E) Production of wood pulp by United States companies will not increase sharply during this year.

Question Discussion & Explanation

**Correct Answer** - B - (click and drag your mouse to see the answer)

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Gold depreciated at a rate of per year between 2000 and 2005. If 1 kg of gold cost dollars in 2001 and dollars in 2003, how much did it cost in 2002 in terms of and ?

A.

B.

C.

D.

E.

Question Discussion & Explanation (Click and drag your mouse to see the correct answer)

**Correct Answer** - E -

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**Verbal (CR)**

Studies have shown that **people who keep daily diet records are far more successful at losing weight than people who don’t keep track of what they eat.** Researchers believe that many weight-loss efforts fail because people eat more calories than they intend to consume. One study followed a group of patients who reported that they could not lose weight when consuming only 1,200 calories a day. **The study found that the group consumed, on average, 47% more than it claimed and exercised 51% less.** In contrast, when dieters record what they eat, their actual consumption more closely matches their reported consumption.

The two boldface portions in the argument above are best described by which of the following statements?

(A) The first is a conclusion reached by researchers; the second is evidence that that conclusion is correct.

(B) The first is an explanation of why a certain theory is thought to be true; the second is an example of research results that support this theory.

(C) The first is an example illustrating the truth of a certain theory; the second is a competing theory.

(D) The first is a premise upon which the researchers base their opinion; the second illustrates that their opinion is correct.

(E) The first introduces a theory that the researchers have disproved; the second is the basis for the researchers’ argument.

Question Discussion & Explanation

**Correct Answer** - D - (click and drag your mouse to see the answer)

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What is the product of three consecutive integers?

(1) At least one of the integers is positive.

(2) The sum of the integers is less than 6.

Question Discussion & Explanation

**Correct Answer** - C - (click and drag your mouse to see the answer)

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**Verbal (SC)**

Shipwrecks are more likely to be found undisturbed at great depths than in shallow coastal waters, which exposes archaeological remains to turbulence and makes them accessible to anyone in scuba gear, whether they be archaeologist, treasure hunter, or sport diver.

(A) than in shallow coastal waters, which exposes archaeological remains to turbulence and makes them accessible to anyone in scuba gear, whether they be

(B) than in shallow coastal waters, where archaeological remains are exposed to turbulence and are accessible to anyone in scuba gear, whether

(C) as opposed to shallow waters along the coast, where archaeological remains are exposed to turbulence and accessible to anyone in scuba gear, including

(D) instead of in shallow waters along the coast, which exposes archaeological remains to turbulence and making them accessible to anyone in scuba gear, including an

(E) instead of shallow coastal waters, because it exposes archaeological remains to turbulence and make them accessible to anyone in scuba gear, whether

Question Discussion & Explanation

**Correct Answer** - B - (click and drag your mouse to see the answer)

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