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9/2/2015 Mrunal [Aptitude] LCM, HCF, GCD: Basic concept, calculation, applications explained Mrunal

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[Aptitude] LCM, HCF, GCD: Basic concept, calculation,applications explained

Posted BySupport StaffOn 17/03/2013 @ 12:22 pm In Aptitude | 127 Comments

1. Introduction2. What is Prime number?3. What is LCM?

LCM4 EXamHow to find LCM using PrimeFactorization?LCM of two numbers (56, 96)LCM of three numbers: (12,15,20)LCM of prime numbersLCM of coprime numbers

4. What is HCF or GCD?HCF finding: Prime FactorizationHCF of two numbers (4, 6)HCF of three numbers (12,24,36)HCF of prime numbers (13,29)HCF of coprime numbers (12,25)HCF vs LCM: #1 multiplicationHCF vs LCM: #2 MagnitudeHCF vs LCM: of fractions

5. for more practice on LCM, HCF

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 forexample1 x 12=12.2 x 6 =123 x 4=12That 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 Nonprime (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 multiple4 x 1 = 4 6 x 1 = 64 x 2 = 8 6 x 2 = 124 x 3 = 12 6 x 3 = 184 x 4 = 16 6 x 4 = 244 x 5 = 20 6 x 5 = 304 x 6 = 24 6 x 6 = 364 x 7 = 28 6 x 7 = 424 x 8 = 32 6 x 8 = 484 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 = 124 x 6 = 24 6 x 4 = 244 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 thesemultiples? 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 oftwo or three numbers? There are many tricks, the easiest one is primefactorization. We’ll learnthat 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 Jethalaltakes 6 minutes to finish it. Now both of them start running from the same point at the sametime in the same direction. They’ll continue running on this track forever. So after how manyminutes 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 timespeeddistance/work, pipescistern etc questions, is already

covered in old articles. (Mrunal.org/aptitude)

How to find LCM using PrimeFactorization?

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

Number Factors46

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

Number Factors4 2 x 26 2 x 3

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

Number factors4 22

6 2 x 3

Now make the third row called “LCM”.

Number factors4 22

6 2 x 3LCM

Now write all prime numbers in this “LCM row”

Number factors4 22

6 2 x 3LCM 2, 3

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Write maximum power of each prime number

Number factors4 22

6 2 x 3LCM 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 factors4 22

6 2 x 3LCM 22 x 3 =12

That’s our answer. LCM (4,6)=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 Factors5696



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

Number Factors56 8 x 796 12 x 8

but we need factors in “prime number” format. 12 and 8 are not prime numbers. So let’sSimplify 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 yourhead, if you start making every calculation on a piece of paper, you’ll run out of time in theexam).

Number Factors56 23 x 796 25x3

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

Number Factors56 23 x 796 25x3LCM 2 3 7

Now write maximum powers of each prime number.

Number Factors56 23 x 796 25x3LCM 25 3 7

Multiply these numbers

Number Factors56 23 x 796 25x3LCM 25 x3x7=32×21=672

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

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



Approach is same. Make prime factors

Number Prime factors12 22 x 315 3 x 520 22 x 5

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

Number Prime factors12 22 x 315 3 x 520 22 x 5LCM 2,3,5

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

Number Prime factors12 22 x 315 3 x 520 22 x 5LCM 22, 3, 5

Now multiple the numbers in last row

Number Prime factors12 22 x 315 3 x 520 22 x 5LCM 22x3x5=60

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

12 x 5 = 6015 x 4 = 6020 x 3 = 60.

So 60 is the least common multiple.

LCM of prime numbers



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

Numbers Factors7 7 x 111 11 x 113 13 x 1LCM 1x 7 x 11 x 13 =1001

So 1001 is the answer.

LCM of coprime numbers

Co prime numbers are those numbers that donot have any common factors. For example, 14and 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 coprime numbers(when they’re given together).Any two consecutive numbers are coprime numbers. (e.g. 11,12 or 1548,1549).In case of coprime numbers, just multiply them and you will get LCM. There is no need tofind factors. example

6 2 x 37 7LCM 2 x 3 x 7 = (6)x7 =42

Advantages of this method?

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, pipecistern 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.

Disadvantages?

3. Becomes tedious, as the number grows bigger, for example LCM (235, 512). There are othermethods to solve those LCMs, but let’s not complicate this article any further. Let’s stick tothis PrimeFactorization 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 = 31 x 2 = 2 2 x 2 = 4 3 x 2 = 61 x 3 = 3 2 x 3 = 6 3 x 3 = 91 x 4 = 4 2 x 4 = 8 3 x 4 = 121 x 5 = 5 2 x 5 = 10 3 x 5 = 151 x 6 = 6 2 x 6 = 12 3 x 6 = 181 x 7 = 7 2 x 7 = 14 3 x 7 = 211 x 8 = 8 2 x 8 = 16 3 x 8 = 241 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=41 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. Inother 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 zeroremainder. In other words, 2 is the factor of both 4 and 6. In other words, 4 and 6 come in thetable 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 thisfield. 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 thebiggest 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 shecan 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! Sohere is the shortcut technique. We’ll use the same approach we’ve used in LCM method: primefactorization.

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 3HCF 21

Therefore, HCF (4,6)=2If 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 624 3 x 836 6 x 6



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

12 2 x 2 x 3=22 x 324 3 x 2 x 2 x 2=23 x 336 3 x 2 x 3 x 2=22 x 32

In the exam you’ve to do this in your ^head.

12 22 x 324 23 x 336 22 x 32

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

12 22 x 324 23 x 336 22 x 32

HCF 22x3

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

12 22 x 324 22 x 2 x 336 22 x 3 x 3HCF 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 theclarity let’s do it

13 13 x 129 29 x 1HCF 1 (because 1 is common in both)

HCF of coprime 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 thenHCF 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 givennumbers)

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. Forexample

Numbers LCM12,15,20 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 HCF12,15,20 1 so it is smaller than smallest number 1215,30 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. Inthe mean time, try finding LCM and HCFs of following numbers



Question Answer (LCM, HCF)91, 12 1092, 146, 69 138, 2369, 97 6693, 163, 33 693, 372, 58 2088, 25, 84 420, 191, 41 3731, 165, 57 3705, 174, 12 444, 244, 55 220, 118, 28, 175 1400, 1

LCM, HCF of fractions

Just observe the color pattern in following image:

for more practice on LCM, HCFBook Chapter no.Quantitative Aptitude, R.S.Agarwal 2Fast track Arithmetic, Rajesh Verma 2Quantam CAT, Sarvesh Kumar Ex.1.3, 1.4Arun Sharma (CAT) 1

In all such books, the authors first give 56 illustration examples and then exercises. I suggest yousolve the the illustration examples as well. After all aptitude is all about practice.



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