### Book 4 Aptitude?

Essential book for Reasoning portion of SSC CGL, SBI PO, LIC AAO. Buy this from flipkart now!

# Introduction

1. Concept of LCM, HCF important for number theory and remainder based problems (generally asked in SSC CGL, CAT.)
2. LCM is important for time and speed, time and work problems.
3. LCM is also important for circular racetracks, bells, blinking lights, etc.
4. HCF is important for largest size of tiles, largest size of tape to measure a land etc.

But before getting into LCM, HCF, let’s understand

# What is Prime number?

• Consider this number : 12. This number can be found in many multiplication tables for example
• 1 x 12=12.
• 2 x 6 =12
• 3 x 4=12
• That means, 12 has many factors (1,2,3,4,6,12). Such number is called a composite number.
• On the other hand, consider this number: 29. You cannot find it in any table except 29 x 1 =29. Such number is called a prime number.
• Let’s make a shortlist from exam point of view
 Prime Non-prime (composite) 2,3,5,7,11,13,17,19,23,29 4,6,8,9,10,12,14,15….

Now hold this prime number thought in your mind for a while.

# What is LCM?

First, let’s create multiplication tables of 4 and 6.

 4′s table multiple 6′s table multiple 4 x 1 = 4 6 x 1 = 6 4 x 2 = 8 6 x 2 = 12 4 x 3 = 12 6 x 3 = 18 4 x 4 = 16 6 x 4 = 24 4 x 5 = 20 6 x 5 = 30 4 x 6 = 24 6 x 6 = 36 4 x 7 = 28 6 x 7 = 42 4 x 8 = 32 6 x 8 = 48 4 x 9 = 36 6 x 9 = 54
• Do you see any common numbers in the multiples of 4 and 6?
• Yes I see 12, 24 and 36 are common in both tables. Let’s isolate them.
 4 x 3 = 12 6 x 2 = 12 4 x 6 = 24 6 x 4 = 24 4 x 9 = 36 6 x 6 = 36
• Ok so 12, 24 and 36 are common multiples of 4 and 6. But what is the smallest of these multiples? Ans 12 is smallest.

In the exam, we’ve no time to make such ^big tables to find LCM. So how to quickly find LCM of two or three numbers? There are many tricks, the easiest one is prime-factorization. We’ll learn that in a bit, but before that:

# LCM4 EXam

1. Suppose there is a circular race track. Tarak Mehta takes 4 minutes to finish it and Jethalal takes 6 minutes to finish it. Now both of them start running from the same point at the same time in the same direction. They’ll continue running on this track forever. So after how many minutes will they meet for the first time on the starting point? Ans. LCM of time = LCM (4,6)=12 minutes. They’ll meet again on the starting point after 12 minutes.
2. Two bells ring at an interval of 4 and 6 minutes respectively. After how many minutes will they ring together? Ans LCM (4,6)
3. Two traffic lights blink at an interval of 40 and 60 seconds respectively. After how many minutes will they link together? Ans LCM (40,60).
4. HCF is also important for remainder related questions. but I’ll cover that in a separate article.
5. How to apply LCM in time-speed-distance/work, pipes-cistern etc questions, is already covered in old articles. (Mrunal.org/aptitude)

# How to find LCM using Prime-Factorization?

Suppose in the exam, we need to find LCM of 4 and 6.
Make a table like this

 Number Factors 4 6

Now you need to find the prime factors of 4 and 6.

 Number Factors 4 2 x 2 6 2 x 3

Express it in terms of “powers”. For example 2 x 2 =22

 Number factors 4 22 6 2 x 3

Now make the third row called “LCM”.

 Number factors 4 22 6 2 x 3 LCM

Now write all prime numbers in this “LCM row”

 Number factors 4 22 6 2 x 3 LCM 2, 3

Write maximum power of each prime number

 Number factors 4 22 6 2 x 3 LCM 22, 3

As you can see, maximum power of 2 was 22 (in 4’s row).
Now multiple the numbers given in LCM row

 Number factors 4 22 6 2 x 3 LCM 22 x 3 =12

If I plot this LCM situation on a Venn Diagram, it’ll look like this:

Anyways, Let’s try a difficult one: 56 and 96.

# LCM of two numbers (56, 96)

 Numbers Factors 56 96

First recall, in which tables do they come? Well 56 comes in 8’s table and 96 comes in 12’s table.

 Number Factors 56 8 x 7 96 12 x 8
• but we need factors in “prime number” format. 12 and 8 are not prime numbers. So let’s Simplify further.
• 56 = 8 x 7 = 23 x 7 (; because 8 = 4 x 2 = 2 x 2 x 2)
• 96 = 12 x 8 = (4×3)x(4×2)=( 22x3) (23)=25x3 (please note you have to do this things in your head, if you start making every calculation on a piece of paper, you’ll run out of time in the exam).
 Number Factors 56 23 x 7 96 25x3

Now let’s make the LCM row. Write all prime numbers (2,3 and7) in ascending order.

 Number Factors 56 23 x 7 96 25x3 LCM 2 3 7

Now write maximum powers of each prime number.

 Number Factors 56 23 x 7 96 25x3 LCM 25 3 7

Multiply these numbers

 Number Factors 56 23 x 7 96 25x3 LCM 25 x3x7=32×21=672

So LCM (56,96)=672
let’s try finding LCM of three numbers.

# LCM of three numbers: (12,15,20)

Approach is same. Make prime factors

 Number Prime factors 12 22 x 3 15 3 x 5 20 22 x 5

Make a new row, write all prime factors in ascending order.

 Number Prime factors 12 22 x 3 15 3 x 5 20 22 x 5 LCM 2,3,5

In the last row, Write the maximum power of those prime numbers.

 Number Prime factors 12 22 x 3 15 3 x 5 20 22 x 5 LCM 22, 3, 5

Now multiple the numbers in last row

 Number Prime factors 12 22 x 3 15 3 x 5 20 22 x 5 LCM 22x3x5=60

Therefore LCM (12,15,20)=60.
You can also look at it in following way:

• 12 x 5 = 60
• 15 x 4 = 60
• 20 x 3 = 60.

So 60 is the least common multiple.

# LCM of prime numbers

Find LCM of 7,11,13
We already know these are prime numbers. So they’ll not have any common factors. We just have to multiply them together and we’ll get LCM. But for the sake of conceptual clarity

 Numbers Factors 7 7 x 1 11 11 x 1 13 13 x 1 LCM 1x 7 x 11 x 13 =1001

# LCM of co-prime numbers

• Co prime numbers are those numbers that donot have any common factors. For example, 14 and 15.
• Individually none of them is prime number because 14=2 x 7 and 15 = 3 x 5.
• But they (14 and 15) donot have any common factors. So they’re called co-prime numbers (when they’re given together).
• Any two consecutive numbers are co-prime numbers. (e.g. 11,12 or 1548,1549).
• In case of co-prime numbers, just multiply them and you will get LCM. There is no need to find factors. example
 6 2 x 3 7 7 LCM 2 x 3 x 7 = (6)x7 =42

1. Extremely fast when you’ve to find LCMs of two digit numbers for example 12,15,96.
2. And usually in time speed work, pipe-cistern type questions have number in two digits (e.g. 12, 15, 96)…so it is very easy to recall in which multiplication tables do they come.

1. Becomes tedious, as the number grows bigger, for example LCM (235, 512). There are other methods to solve those LCMs, but let’s not complicate this article any further. Let’s stick to this Prime-Factorization method for a while.

Ok so far we know what is LCM and how to find HCF/GCD?

# What is HCF or GCD?

• HCF= Highest common factors.
• GCD= Greatest common divisor. Names are different otherwise they’re one and same.
• Suppose you’ve to find the HCF of (4 and 6).
• I’ll write the tables of numbers that come before 4 and 6 (i.e. 1, 2 and 3.)
 1 x 1 = 1 2 x 1 = 2 3 x 1 = 3 1 x 2 = 2 2 x 2 = 4 3 x 2 = 6 1 x 3 = 3 2 x 3 = 6 3 x 3 = 9 1 x 4 = 4 2 x 4 = 8 3 x 4 = 12 1 x 5 = 5 2 x 5 = 10 3 x 5 = 15 1 x 6 = 6 2 x 6 = 12 3 x 6 = 18 1 x 7 = 7 2 x 7 = 14 3 x 7 = 21 1 x 8 = 8 2 x 8 = 16 3 x 8 = 24 1 x 9 = 9 2 x 9 = 18 3 x 9 = 27

Ok, in which number’s table (1, 2 or 3) do you see both 4 and 6 reappearing?
There are two such tables 1’s table and 2’s table.

 4 and 6 are common in 1’s table. 4 and 6 are common in 2’s table. 1 x 4=4 2 x 2=4 1 x 6=6 2 x 3=6.

## What does ^this mean?

• If I divide 4 by 1, I get zero remainder. Similarly if I divide 6 by 1, I get zero remainder. In other words, 1 is the factor of both 4 and 6. In other words, 4 and 6 come in the table of 1.
• Similarly, If I divide 4 by 2, I get zero remainder. Similarly if I divide 6 by 2, I get zero remainder. In other words, 2 is the factor of both 4 and 6. In other words, 4 and 6 come in the table of 2.
• Thus, 4 and 6 have two common factors (1 and 2) but highest of these common factors is 2. Therefore HCF of (4,6)=2.

## HCF 4 EXAM?

• What is the highest number that’ll divide 4 and 6 evenly. Ans HCF (4,6)
• There is a 4 x 6m rectangular farm. Find the length of longest tape that can measure this field. Ans HCF (4,6)
• There is a 4x 6cm floor. Find the length of largest square tile that can be evenly laid on it. Ans HCF (4,6)
• Two drums contain 400 and 600 liters of desi and foreign liquor respectively. What is the biggest measure (cup) that can measure both of them exactly? Ans. HCF (400, 600).
• A teacher has 40 pens and 60 pencils. Find maximum number of students among whom she can distribute these items evenly.
• HCF is also important for remainder related questions. but I’ll cover that in a separate article.

# HCF finding: Prime Factorization

In the exam, we can’t make multiplication tables of every number preceding the given numbers! So here is the shortcut technique. We’ll use the same approach we’ve used in LCM method: prime factorization.

# HCF of two numbers (4, 6)

First make prime factors of given numbers.

 4 22 6 2 x 3

Now, make third row: HCF and write the prime numbers that are common in both numbers.

 4 22 6 2 x 3 HCF 21

Therefore, HCF (4,6)=2
If I’ve to plot the HCF of 4 and 6 on a Venn diagram, it’ll look like this:

# HCF of three numbers (12,24,36)

 12 2 x 6 24 3 x 8 36 6 x 6

But I want them in prime format. So I’ll further simplify.

 12 2 x 2 x 3=22 x 3 24 3 x 2 x 2 x 2=23 x 3 36 3 x 2 x 3 x 2=22 x 32

 12 22 x 3 24 23 x 3 36 22 x 32

Now make a new row, write the prime numbers that are common in all of above.

 12 22 x 3 24 23 x 3 36 22 x 32 HCF 22x3

^in case you’re confused, let me rewrite and do it again

 12 22 x 3 24 22 x 2 x 3 36 22 x 3 x 3 HCF 22x3

The numbers highlighted in bold are common. Therefore HCF = 22 x 3=12.

# HCF of prime numbers (13,29)

Prime numbers donot have any common factors. So HCF of such numbers is always 1. But for the clarity let’s do it

 13 13 x 1 29 29 x 1 HCF 1 (because 1 is common in both)

# HCF of co-prime numbers (12,25)

Again same: 1, because co prime numbers donot have common factors.
Similarly consecutive numbers (like 456,457) donot have common factors either.
Therefore, in all such cases, HCF =1.

# HCF vs LCM: #1 multiplication

If we’ve two numbers a and b. and their HCF and LCM are given then
HCF x LCM = a x b.
But this relation only work for TWO numbers and not for more than two numbers.
Let’s understand this with an example.
You know that LCM (4,6)=12 and HCF (4,6)=2.

 Left hand side (LCM x HCF) Right hand side (multiplication of given numbers) 12 x 2 4 x 6 =24 =24

So both sides match. Therefore, in case of two numbers (a and b)
LCM X HCF = a x b.
But this is not always true for three numbers. For example, Find LCM and HCF of 12,15,20. You’ll get HCF=1 and LCM=60.

 Left hand side (LCM x HCF) Right hand side (multiplication of given numbers) 60 x 1 12 x 15 x 20 =60 =3600

In this case, both sides donot match.

# HCF vs LCM: #2 Magnitude

For any given numbers, their LCM is always greater than or equal to the biggest number. For example

 Numbers LCM 12,15,20 60 so greater than biggest number (20) 15,30 30. which is equal to the biggest number (30).

Similarly, for HCF, the HCF of given numbers is always less than or equal to the smallest number. For example

 Numbers HCF 12,15,20 1 so it is smaller than smallest number 12 15,30 15. so it is equal to the smallest number 15.

Ok this is just the basic overview. In the next article, we’ll see the application of these concepts. In the mean time, try finding LCM and HCFs of following numbers

 Question Answer (LCM, HCF) 91, 12 1092, 1 46, 69 138, 23 69, 97 6693, 1 63, 33 693, 3 72, 58 2088, 2 5, 84 420, 1 91, 41 3731, 1 65, 57 3705, 1 74, 12 444, 2 44, 55 220, 11 8, 28, 175 1400, 1

# LCM, HCF of fractions

Just observe the color pattern in following image:

# for more practice on LCM, HCF

 Book Chapter no. Quantitative Aptitude, R.S.Agarwal 2 Fast track Arithmetic, Rajesh Verma 2 Quantam CAT, Sarvesh Kumar Ex.1.3, 1.4 Arun Sharma (CAT) 1

In all such books, the authors first give 5-6 illustration examples and then exercises. I suggest you solve the the illustration examples as well. After all aptitude is all about practice.

### 56 comments to [Aptitude] LCM, HCF, GCD: Basic concept, calculation, applications explained

• muana

good…

• Riddhi

its ryl very good n helpful…

• maheshwari

hello sir, can u explain the probability problems.

• Suriya Prakash

Its given in d aptitude section…

• maheshwari

permutation and combinations.

• Siddharth Raj

Sir,I am a B.Sc wtih Zoology+Botany+Chemistry
Which Subject I fill In Colmn…….Subject of graduation
Plz help !!!!!!!!

• siri

hi,even i have the same problem in filling my application as BscBiotechnology was not mentioned in that,should i opt for “other” category,some body sugeest me quickly.

• Neeraj kumar

you have made it so easy.. thanks sir..

• SIDHARTHA

• Hey friends i had talked 2 1 of my friends 2day in Delhi……people of general category want attempts 2 b renewed as the new pattern is a major overhaul made by upsc…it was done in 1979…attempts were renewed when new pattern was introduced….in 1 or 2 days they r goin 2 meet Mr Narayanswami and persuade him and upsc 2 renew all 4 attempts again…so we need a strong support….hope all will support this v good cause….u cn login 2 orkut community 4 this….http://www.orkut.com/Main#Community?cmm=23492063

this is what i got 2 know 4m him…..so i hope all candidates will support this cause

• ASHISH

Thank sir…it’ll help us to solve many questions easily and timely….

• Shri

Hi mrunal Ji,The print option is not good as the old one,please take into consideration! The old allows to resize,remove letters and it needs less page than this new way.Is there anyway to resize the fonts??

• karan

P:

• prem prakash

HELLO MRUNAL SIR
SIR IN THIS ARTICLE U HAVE GIVEN ONLY AN OVERVIEW OF THAT TOPICS. BUT IN SSC CGL THEY ASK DIFFICULT PROBLEMS FROM THIS SECTION. SIR PLZ TAKE A LOOK ON THAT TYPE OF QUESTION ALSO.

• Priyanka Khanna

Hey Mrunal please write articles on reasoning topics like direction,ranking,missing number in figure,mathematical operators,analytical reasoning,clock,and also some non-verbal reasoning topics if you may.
I seriously need tips to solve these topics as the SSC exam is round the corner.
Also,your aptitude topics have been very helpful to me.Keep up the good work! :)

• ravi

respected sir i think that one of the answer of the question is wrong.plz give me clear explanation…………..QUESTION IS There is a 4 x 6m rectangular farm. Find the length of longest tape that can measure this field. Ans HCF (4,6)…………ans should be 2under root 13……..if i m wrong then plz give me right explanation….

thanks

• Keshav

Hi Mrunal,

I had filed an RTI for knowing whether i am covered under OBC or not. But they did not reply so on Feb 19 i filed 2nd appeal which is expected to heard in May or June.

Please tell me whethet i should appear in exam as OBC or General category?
If now i appear for OBC if later on i am not covered under OBC then whether i will be eligiuble for appearance in Mains under General cartegory or not?

If now i appear as General and later on can i appear as an OBC or not?

• mayank

sir ji mujhe bhi bata dijiyega is matter me kuch pata chale toh

• Dc

Same question here also!
Where did u file RTI and whats ur doubt?

• Keshav

I have doubt whether i am covered under OBC or not as some people of my caste has made this certificate but i want to check.

• Gagan

One can check in the list of castes provided by National commission of backward class (NCBC).

• Keshav

• There is no need to file an R.T.I to find out whether you’re OBC or not. It’s straightforward procedure:
To get OBC reservation you need two certificates
1. OBC caste certificate
2. Non-Creamy layer certificate.

To get these certificates, you need to visit local Mamlatdaar/ Tehsildaar/ Collector’s office or Zila Seva Sadan (whatever applicable in your city/town).
so go there, ask questions to concerned officials. They’ll tell you whether your caste is under Central list of OBCs or not. (else check the ministry of social justice’s website)
http://socialjustice.nic.in/pdf/bc081211.pdf

for any doubts on Non-creamy layer, visit obcguru.com

• LUCKY

Sir, pls provide the articles for the remaining topics of trigonometry.

• sravan

sir can you give guidance for sail exam(MT)

• TPK

can any one explain:”Life insurers not allowed to participate in repo transactions: IRDA”

• Rupesh

Great Job for all

Runal Da ,
i think you should gives us a time on skype to personally discuss certain prob with you . you can schedule a time for it ,once in a week …for me you are my guru dronacharya coz i don’t have money to join the coaching so i’m just following your instructions

• ravi

sir please provide theorems list to remember in geometry for ssc …give some articles on geometry also …thankyou

• tracy

thanx alot for showing such a good way of study and i salute u .your material is really worthy.do keep such blessing on us by providing more good and precise material.

• moumita

Sir will plz come up with a video class for MENSURATION part

• VIPIN SHARMA

GRT……………………………..
WEBSITE.

• parmod

fantabulous site thanx a lot working so hard i appreciate your efforts

Its simply great…!

• M RAMANJINEYULU

IT IS SIMPLE AND EASY METHEOD

• Umapathi Krishna

easy to understand :)

• shahid

what an idea sir?

• Mahesh Meena

Thanks a lot ..you have opened all the doors of mind

• Cena

@ prem, which type of difficult problems ? Can u post one here ?

• kannan

its good explaination sir about HCF , LCM

how do you like math math is a good subject Ilike it you like it

hi everyone I am Sharadha ganapathy from 5B of Vidya mandir

• Tanvi

Respected sir,
I am very glad to say that people like you are still there in our country. You are simply awesome sir. Your knowledge is extremely helpful to all the needy students. May god bless you with lots of knowledge bcoz
YE AAPKA GYAN HI HUMHE HUMHARA HAQ DILAYEGA…. thanks a lot for ur support sir

• vishnu

can you explain about the trignometry in simple manner

• Munnaf

thank you very much . . . ,

it is very useful.

• sivachandran

realy very useful web,
much thanks
its like a teacher

Dear Sir,
Its so informative and you have got the talent to express and teach things in a much simpler way..Thanks a lot and I am so glad to see this!

• Liggy

Sir :
Question : A teacher has 40 pens and 60 pencils. Find maximum number of students among whom she can distribute these items evenly.

To distribute the pens and pencils evenly would mean that a student who gets pen , gets a pencil too . Hence , a maximum of 50 students would get a pencil and a pen . This would be even distriution.

Would this be correct ?
Thanks.Liggy

• Vikas Shukla

This is very helpful for new student who want to learn LCM & HCF.

Thanks you very very much.

• Trupti

Its a very easy method and nice examples. I could solve it very quickly. I got to remind of my school days.

• Mushinala Vijaya Kumar

thq sir Given more Maths prblms

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Very Much Sir

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• Chetan Belle

This was a good session. Helped me brush up my memories.
Preparing for GRE. This indeed helped me. Thanks a lot…

• roshan singh

tq so much sir really so helpful