Number Properties e-book - e-GMAT · 2019-09-06 · ©e-GMAT LLC. Unauthorized copying for commercial and competitive purposes is not allowed. Example So, let’s think through the

6/16/2015

Number Properties e-book

By e-GMAT Team E-GMAT LLC

©e-GMAT LLC. Unauthorized copying for commercial and competitive purposes is not allowed.

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

Getting intimidated by complex expressions ......................................................................................................... 4

What we mean ................................................................................................................................................................. 4

What can you do to avoid this pitfall? ................................................................................................................... 4

Takeaway .......................................................................................................................................................................... 6

Test Yourself .................................................................................................................................................................... 7

Wasting time on unimportant terms ........................................................................................................................... 7

What we mean ................................................................................................................................................................. 7

What can you do to avoid this pitfall? ................................................................................................................... 7

Takeaway .......................................................................................................................................................................... 8

Test Yourself .................................................................................................................................................................... 9

Getting stumped in division ............................................................................................................................................ 9

What we mean ................................................................................................................................................................. 9

What can you do to avoid this pitfall? ................................................................................................................. 10

Takeaway ........................................................................................................................................................................ 11

Test Yourself .................................................................................................................................................................. 11

Conclusion ............................................................................................................................................................................ 12

Practice Questions ............................................................................................................................................................ 13

Question 1 ........................................................................................................................................................................ 13

Question 2 ....................................................................................................................................................................... 13

Question 3 ....................................................................................................................................................................... 13

3 deadly mistakes you must avoid in LCM-GCD Questions ................................................................................. 14

Getting confused by different names of LCM-GCD .......................................................................................... 15

Different way to express GCD ................................................................................................................................ 15

Different ways to express LCM ............................................................................................................................. 16

Takeaway ........................................................................................................................................................................ 17

Not having prime factorization as the default approach ................................................................................ 17

Takeaway ........................................................................................................................................................................ 19

Being unmindful of constraints on LCM-GCD ....................................................................................................... 19

Range of GCD ................................................................................................................................................................ 19

Range of LCM ................................................................................................................................................................ 20

Takeaway ........................................................................................................................................................................ 22


Conclusion ............................................................................................................................................................................ 23


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!

http://gmatclub.com/forum/the-e-gmat-number-properties-knockout-great-700-level-ques-195889.html



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 (j3-27)2(j3+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 (j3-27)2(j3+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) (j3 – 27)2 will have the same even-odd nature as (j3 – 27). Similarly, (j3 + 1)3 will have the

same even-odd nature as (j3+1)

ii) j3 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*NK + P where N and K are positive integers, is X divisible by 2?

1. N + KN = 915

2. P35 + 35P is Even

The detailed solution of this question is available here

This is the end of mistake 1. Click here to go back to the table of contents.

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 analyzing

whether a is even or odd, because that is what will get you to the answer.

If you fall into the pitfall of analyzing 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

https://e-gmat.com/blogs/becoming-a-gmat-even-odd-champion-q1/


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 – b3, is n2 + 3 divisible by 2?

(1) a2 – 4b3 – 5 = 0

(2) 3b3 – a2 + 6 = 0

The detailed discussion of this question is available here


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:

http://gmatclub.com/forum/if-a-b-and-n-are-positive-integers-such-that-n-3a-b3-195919.html


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:

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:

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):

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 x4 y3 = z2, is x9 – y6 odd?

1. 𝐱𝟒 𝐲𝟑

𝐱𝟐 + 𝐲𝟐 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


http://gmatclub.com/forum/if-x-y-and-z-are-positive-integers-such-that-x-4-y-3-z-196033.html


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!

https://e-gmat.com/registration/?channel=qgcf


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 = A3 -1

(2) A is a product of two consecutive prime numbers and when 𝑩 + 𝟑𝟏𝟏 is added to A, the

sum is an odd number.


Question 2

If P and Q are positive integers, is the product 3PQ divisible by 2?

1. 6Q3 + 2 is an even number

2. P + 8Q2 is a prime number


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. a3*(c-1)2 is odd


This is the end of the article “Do you make these 3 mistakes in GMAT Even-Odd Questions?” Click here

to go back to the table of contents.

http://gmatclub.com/forum/is-the-product-of-two-integers-a-and-b-odd-195888.html




3 deadly mistakes you must avoid in LCM-

GCD Questions

Before you start reading this article, ask yourselves the following question:

a) Are you among those who wish that they do not encounter a LCM-GCD question on their exam

day or

b) Are you among those who take a guess whenever they come across a LCM-GCD question or

c) Are you among those who try their mighty hard but still fall short of the correct answer in a

LCM-GCD question

If yes, this article was waiting for you the moment you thought of taking the GMAT. We at e-GMAT

believe in making GMAT ‘easy’ for you. This piece of article is one among the many in that direction.

Based on our extensive student interaction for more than 1000+ students spread over the live

sessions, forum queries and our recently conducted The E-GMAT Number Properties Knockout that

was attempted more than 5000 times, we have identified these three primary pitfalls in the concepts

of LCM-GCD:

1. Getting confused by different names of LCM & GCD

2. Not having prime factorization as the default approach

3. Being unmindful of constraints on LCM & GCD

In this article, we will explain the pitfalls in LCM-GCD questions along with the measures to overcome

these pitfalls to help you ace 700+ questions of the topic. Let’s begin with the first and the most

common pitfall.



Getting confused by different names of LCM-GCD

LCM-GCD are defined in various terms and phrases in questions which appear in GMAT. If you are

not able to decipher the hidden meaning behind these terms you are likely to waste your precious

time and even worse may approach the question with a disastrous approach. Thus, it is of utmost

importance to know when the question is talking about LCM-GCD. Let’s see the ways in which a

question can encrypt the concept of LCM-GCD.

Different way to express GCD GCD can be expressed as the following terms/phrases:

a) x is the GCD of 12 & 18

b) x is the HCF of 12 & 18 (HCF stands for Highest Common Factor)

c) x is the highest number which divides both 12 & 18

All the 3 terms above refer to the GCD of 12 & 18 i.e. 6. Try focusing on keywords such as ‘Highest’,

‘Greatest’. These words are an indication of the question talking about the GCD of a set of numbers in

a LCM-GCD question.

Let’s see now the terms/phrases which are used to express LCM:


Different ways to express LCM

a) z is the LCM of 12 & 18

b) z is the lowest number which is divisible by both 12 & 18

c) z is the lowest number which has 12 & 18 as its factors

All the above terms refer to the LCM of 12 & 18 i.e. 36. Try focusing on keyword ‘lowest’ which can

be an indication of question talking about LCM of a set of numbers.

The important point to remember for identification of both LCM & GCD is the definition of these

terms.

While GCD is the factor/divisor of a set of numbers, LCM is the multiple of a set of numbers.

Whenever you are presented with a phrase which confuses you between LCM-GCD, try to breakdown

the statement in the terms of factors and multiples. For example, refer the following question:

P1.1 Two positive integers a and b are divisible by 5, which is their largest common

factor. What is the value of a and b?

I. The lowest number that has both integers a and b as its factors is the

product of one of the integers and the greatest common divisor of the two

integers.

II. The smaller integer is divisible by 4 numbers and has the smallest odd

prime number as its factor.


The question plays with words when talking about LCM-GCD. Try answering this question by

deciphering the message of LCM-GCD hidden in it. You can access the detailed solution here.

Takeaway I. LCM of a set of numbers is divisible by all the numbers in the set

II. GCD of a set of numbers divides all the numbers in the set

Keeping these two definitions in mind and your heart will always keep you in good stead whenever

you face a LCM-GCD question.


Not having prime factorization as the default approach

Many of the students at some time or the other have avoided doing prime factorization of a set of

numbers for finding their LCM & GCD. Such an approach towards a LCM-GCD question most often

leads to incorrect solution costing you precious marks and time

We will explain you here, why and how prime factorization is a robust method in LCM-GCD questions

and the potential pitfalls which you may avoid on your journey to a perfect quant score

Let’s look at the following official question:

P2.1 If n is a positive integer, is n a multiple of 24?

I. n is a multiple of 4

http://gmatclub.com/forum/two-positive-integers-a-and-b-are-divisible-by-5-which-is-their-195951.html


II. n is a multiple of 6

The most common fallacy committed by students in such a question is to express n as a multiple of 4

& 6 i.e. 4x and 6y respectively and write n = 24xy. With this they conclude that n is a multiple of 24

which is an incorrect analysis.

Students who get it right are the ones who prime factorize 4 & 6 and find that

I. n is a multiple of 22 - so, n is a number of the form 22*k

II. n is a multiple of 2*3 - so, n is a number of the form 2*3m

The key here is to observe that n can be expressed as a multiple of maximum of 22 and as a multiple

of maximum of 31. Combining these two, we can say that n can be expressed as a multiple of 22 * 3 i.e.

12. Hence n is a multiple of 12 and not 24. Thus it can be seen that the method of prime factorization

helps in eliminating such mistakes in the approach of the students.

The other and the most significant benefit of prime factorization is to help you remember an easy

mnemonic for calculating LCM & GCD. For calculating the GCD (Greatest Common Divisor), use the

lowest power of primes of the set of numbers and for calculating the LCM (Least Common Multiple)

use highest power of primes of the set of numbers. It’s a golden approach where you are never going

to commit an error while calculating LCM-GCD of a set of numbers. Also, this approach is particularly

helpful when calculating LCM-GCD of more than 2 numbers where the division method usually fails

or takes up a lot of time.

Let’s understand another advantage of using prime factorization in LCM-GCD question with an

example:

P2.2 If the LCM of 36 & x is 72, what are the possible values of x?

At the first look, the question seems to have insufficient information provided. Let’s see if the prime

factorization method leads to an answer.

It’s given that 72 is the LCM of 36 & x. Prime factorizing 72 would give us 72 = 23 * 32

Similarly 36 can be prime factorized as 36 = 22 * 32. Now, we know that for calculating LCM we take

the highest power of primes. Since 72 has 32 in it x can have either {30, 31 , 32}. Similarly x will have 23

in it as 36 has only 22 in it and the LCM has 23 in it. Now, x can’t have any more prime factor other

than 2 or 3 as the LCM does not have any other prime factor apart from 2 and 3. So, possible values

of x = {23 * 30, 23 * 31, 23 * 32} = {8, 24, 72}.


Thus, you see that prime factorization leads you out of a situation where the information seemed

insufficient at the very first instant.

Takeaway Use Prime Factorization as your first line of offense against LCM-GCD questions


Being unmindful of constraints on LCM-GCD

A few questions on GMAT test your understanding of the possible values of LCM & GCD of a set of

integers. For example, have a look at the following official question (OG 13 QR2, Problem Solving

QNo: 98):

P3.1 Which of the following CANNOT be the greatest common divisor of two positive

integers x and y?

A. 1

B. x

C. y

D. x – y

E. x + y

The above question directly tests your knowledge of the range of LCM & GCD. Students not aware of

the range of GCD would have trouble solving this question in the given time frame. The solution to

this question can be accessed here

Let’s see & observe how we can find the range of LCM & GCD and how it can be used for solving

questions on LCM-GCD.

Range of GCD As we know, GCD of a set of integers is the factor of all the integers in the set. Now, factor of an integer

can’t be greater than the magnitude of the integer itself. Thus, the GCD of a set of integers can’t be

greater than the magnitude of the smallest integer of the set.

GCD of a set of integers < = Magnitude of the smallest integer of the set

https://e-gmat.com/blogs/3-mistakes-of-lcm-gcd-questions-q4/


Also, we know that 1 is the factor of all integers. Thus, every integer will have at least one common

factor i.e. 1. So, for integers which do not have any common prime factors, their only and hence their

highest common factor would be 1.

GCD of a set of integers => 1

From the above two analysis we can conclude that

1 < = GCD of a set of integers < = Magnitude of the smallest integer of the set

Let’s see the range of LCM now.

Range of LCM As we know, LCM of a set of integers is the multiple of all the integers in the set. Now, the multiple of

an integer can’t be smaller than the magnitude of the integer itself. Hence, common multiple of a set

of integers can’t be smaller than the magnitude of the largest integer of the set.

LCM of a set of integers => Magnitude of the largest integer of the set

For finding the upper range of LCM of a set of integers let’s assume two integers

P = ax * by and Q = cz where a, b, c are three distinct prime numbers. As we know that for calculating

LCM we take the highest powers of the prime numbers, so LCM (P, Q) = ax * by * cz = P * Q.

LCM of a set of integers <= Product of magnitude of all integers in the set

From the above two analysis we can conclude that,

Magnitude of the largest integer of the set <= LCM of a set of integers<= Product of magnitude

of all integers in the set


From the above interpretation and knowing that Product of all integers in the set = GCD * LCM, we

can say that when LCM is equal to the product of all integers in the set, GCD = 1. We also remember

that GCD = 1 when the integers of a set do not have any common prime factor. Thus we can say that

for a set of integers which don’t have a common prime factor, GCD = 1 and LCM = Product of all

integers on the set.

Given below are some practice questions which test your concepts of LCM-GCD

P3.2 If A and B are positive integers greater than 1 such that the GCD of A and B is

1, then which of the following must be true?

I. A and B are prime numbers.

II. A and B are consecutive numbers.

III. A and B do not have a common prime factor

IV. The product AB has two prime factors

V. A and B have the opposite even-odd nature

A. I and II only

B. III only

C. I, V only

D. I,II, III, V only

E. I, II, III, IV & V

The solution of the question can be accessed here



P3.3 For two positive integers A & B, what is the highest number that divides

completely the product of integers from 1 to A and 1 to B such that B = A + 29.

A. 1

B. Product of all integers from 1 to A

C. Product of all integers from 1 to B

D. 29*A

E. Can’t be determined

The solution of the question can be accessed here

Takeaway I. 1 < = GCD of a set of integers < = Magnitude of the smallest integer of the set

II. Magnitude of the largest integer of the set <= LCM of a set of integers<= Product of magnitude

of all integers in the set




Conclusion

We have observed from the above pitfalls that it’s easy to make LCM-GCD your friend. Since questions

on LCM-GCD constitute an important part of Number Properties and are frequently tested on GMAT,

it’s advisable to be proficient on this topic. Once you have gone through the above article and

identified the potential pitfalls in LCM-GCD question, practice more questions on such concepts to

prepare yourself better for the exam day.

Hope that this article would be a ‘factor’ in your quant preparation and your joy is a ‘multiple’ of your

quant score.

Wish you a great GMAT journey

This is the end of the article “3 deadly mistakes you must avoid in LCM-GCD questions” Click here to go

back to the table of contents.

Number Properties e-book - e-GMAT · 2019-09-06 · ©e-GMAT LLC. Unauthorized copying for commercial and competitive purposes is not allowed. Example So, let’s think through the

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