Do you make these 3 mistakes in GMAT Even-Odd Questions?

Do you make these 3 mistakes in GMAT Even-Odd Questions?




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.

What we have observed is that there are three primary pitfalls that students fall into in Even-Odd questions:

1. Getting intimidated by complex expressions
2. Wasting time on unimportant terms
3. 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!

1. Getting intimidated by complex expressions

What we mean

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:

i) Either leave the question without answering
ii) 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 a3 + b3 on the terms of this expression, and then, realize, much to your dismay, that you’ve complicated the question even further

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

What can you do to avoid this 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.



Example

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

1st 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,
i) \((j^3-27)^2\) will have the same even-odd nature as \((j^3-27)\). Similarly, \((j^3+1)^3\) will have the same even-odd nature as \((j^3+1)\)
ii) \(j^3\) 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)

2nd 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?


Takeaway

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.


Test Yourself

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)


2. Wasting time on unimportant terms


What we mean

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.


What can you do to avoid this pitfall?

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:


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


Example

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?

1st Pointer

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

2nd 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

3rd 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.

Takeaway

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.



Test Yourself

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. \(a^2\) – 4\(b^3\) – 5 = 0
2. 3\(b^3\) – \(a^2\) + 6 = 0

(The detailed discussion of this question is available here)


3. Getting stumped in division

What we mean

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 \(\frac{XB}{(4A+1})\) 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:

i) Either give up this question as too difficult
ii) 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) is odd. This approach is time-consuming and error-prone, because you may miss out on some possible cases.

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.


What can you do to avoid this pitfall?

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

Convert the division equation into a multiplication equation.

Example

Let’s illustrate this approach on question P3.1

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

\(\frac{X}{B}\)=Even number

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):

\(\frac{(XB}{(4A+1)})\)=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.

Takeaway

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

Test Yourself

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.\(\frac{(x^{4}y^{3})}{(x^2 + y^2 )}\) 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

Conclusion

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!

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Practice Questions

Question 1

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 \(B+3^{11}\) is added to A, the sum is an odd number.

(The detailed solution of this question is available here)

Question 2

If P and Q are positive integers, is the product 3P^Q divisible by 2?
1. 6\(Q^3\) + 2 is an even number
2. P + 8\(Q^2\) is a prime number

(The detailed solution of this question is available here )

Question 3

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)