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Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by Total Gadha - Thursday, 25 December 2008, 01:14 AM
 

What is highest common factor (HCF) and least common multiple (LCM)? How do you calculate HCF and LCM of two or more numbers? Are you looking for problems on HCF and LCM? This chapter will answer all these questions.

HIGHEST COMMON FACTOR (HCF)

The largest number that divides two or more given numbers is called the highest common factor (HCF) of those numbers. There are two methods to find HCF of the given numbers:

Prime Factorization Method- When a number is written as the product of prime numbers, the factorization is called the prime factorization of that number. For example, 72 = 2 × 2 × 2 × 3 × 3 = 23 × 32

To find the HCF of given numbers by this method, we perform the prime factorization of all the numbers and then check for the common prime factors. For every prime factor common to all the numbers, we choose the least index of that prime factor among the given number. The HCF is product of all such prime factors with their respective least indices.

EXAMPLE

Find the HCF of 72, 288, and 1080
Answer:
72 = 23
× 32,
288 = 25
× 32,
1080 = 23
× 33 × 5

The prime factors common to all the numbers are 2 and 3. The lowest indices of 2 and 3 in the given numbers are 3 and 2 respectively.
Hence, HCF = 23
× 32 = 72.

Find the HCF of 36x3y2 and 24x4y.
Answer:
36x3y2 = 22
× 32 × x3 × y2
24x4y = 23
× 3 × x4 × y.

The least index of 2, 3, x and y in the numbers are 2, 1, 3 and 1 respectively.
Hence the HCF = 22
× 3 × x2 × y = 12x2y.

Division method- To find HCF of two numbers by division method, we divide the higher number by the lower number. Then we divide the lower number by the first remainder, the first remainder by the second remainder... and so on, till the remainder is 0. The last divisor is the required HCF.

EXAMPLE

Find the HCF of 288 and 1080 by the division method.

Answer:

                                   

Hence,  the last divisor 72 is the HCF of 288 and 1080.

CONCEPT OF CO-PRIME NUMBERS: Two numbers are co-prime to each other if they have no common factor except 1. For example, 15 and 32, 16 and 5, 8 and 27 are the pairs of co-prime numbers. If the HCF of two numbers N1 and N2 be H, then, the numbers left after dividing N1 and N2 by H are co-prime to each other.

Therefore, if the HCF of two numbers be A, the numbers can be written as Ax and Ay, where x and y will be co-prime to each other.

SOLVED PROBLEMS ON HCF

Three company of soldiers containing 120, 192, and 144 soldiers are to be broken down into smaller groups such that each group contains soldiers from one company only and all the groups have equal number of soldiers. What is the least number of total groups formed?

Answer: The least number of groups will be formed when each group has number of soldiers equal to the HCF. The HCF of 120, 192 and 144 is 24. Therefore, the numbers of groups formed for the three companies will be 5, 8, and 6, respectively. Therefore, the least number of total groups formed = 5 + 8 + 6 = 19.

The numbers 2604, 1020 and 4812 when divided by a number N give the same remainder of 12. Find the highest such number N.

Answer: Since all the numbers give a remainder of 12 when divided by N, hence (2604 - 12), (1020 - 12) and (4812 - 12) are all divisible by N. Hence, N is the HCF of 2592, 1008 and 4800. Now 2592 = 25 × 34, 1008 = 24 × 32 × 7 and 4800 = 26 × 3 × 52. Hence, the number N = HCF = 24 × 3 = 48.

The numbers 400, 536 and 645, when divided by a number N, give the remainders of 22, 23 and 24 respectively. Find the greatest such number N.

Answer: N will be the HCF of (400 - 22), (536 - 23) and (645 - 24). Hence, N will be the HCF of 378, 513 and 621. -->  N = 27.

The HCF of two numbers is 12 and their sum is 288. How many pairs of such numbers are possible?

Answer: If the HCF if 12, the numbers can be written as 12x and 12y, where x and y are co-prime to each other. Therefore, 12x + 12y = 288 --> x + y = 24.

The pair of numbers that are co-prime to each other and sum up to 24 are (1, 23), (5, 19), (7, 17) and (11, 13). Hence, only four pairs of such numbers are possible. The numbers are (12, 276), (60, 228), (84, 204) and (132, 156).

The HCF of two numbers is 12 and their product is 31104. How many such numbers are possible?

Answer: Let the numbers be 12x and 12y, where x and y are co-prime to each other. Therefore, 12x × 12y = 31104 --> xy = 216. Now we need to find co-prime pairs whose product is 216.

216 = 23 × 33. Therefore, the co-prime pairs will be (1, 216) and (8, 27). Therefore, only two such numbers are possible.

LEAST COMMON MULTIPLE (LCM)

The least common multiple (LCM) of two or more numbers is the lowest number which is divisible by all the given numbers.

To calculate the LCM of two or more numbers, we use the following two methods:

Prime Factorization Method: After performing the prime factorization of the numbers, i.e. breaking the numbers into product of prime numbers, we find the highest index, among the given numbers, of all the prime numbers. The LCM is the product of all these prime numbers with their respective highest indices.

EXAMPLE

Find the LCM of 72, 288 and 1080.

Answer: 72 = 23 × 32, 288 = 25 × 32, 1080 = 23 × 33 × 5

The prime numbers present are 2, 3 and 5. The highest indices (powers) of 2, 3 and 5 are 5, 3 and 1, respectively.

Hence the LCM = 25 × 33 × 5 = 4320.

Find the LCM of 36x3y2 and 24x4y.

Answer: 36x3y2 = 22 × 32 × x3 × y2 24x4y = 23 × 3 × x4 × y.
The highest indices of 2, 3, x and y are 3, 2, 4 and 2 respectively.
Hence, the LCM = 23
× 32 × x4 × y2 = 72x4y2.
 

Division Method: To find the LCM of 72, 196 and 240, we use the division method in the following way:

                                               

L.C.M. of the given numbers = product of divisors and the remaining numbers
= 2
× 2 × 2 × 3 × 3 × 10 × 49 = 72 × 10 × 49 = 35280.


PROPERTIES OF HCF AND LCM

·          The HCF of two or more numbers is smaller than or equal to the smallest of those numbers.
·          The LCM of two or more numbers is greater than or equal to the largest of those numbers
·          If numbers N1, N2, N3, N4 etc. give remainders R1, R2, R3, R4, respectively, when divided by the same number P, then P is the HCF of (N1 - R1), (N2 - R2), (N3 - R3), (N4 - R4) etc.
·          If the HCF of numbers N1, N2, N3 ... is H, then N1, N2, N3... can be written as multiples of H (Hx, Hy, Hz.. ). Since the HCF divides all the numbers, every number will be a multiple of the HCF.
·          If the HCF of two numbers N1 and N2 is H, then, the numbers (N1 + N2) and (N1 - N2) are also divisible by H. Let N1 = Hx and N2 = Hy, since the numbers will be multiples of H. Then, N1 + N2 = Hx + Hy = H(x + y), and N1 - N2 = Hx - Hy = H(x - y). Hence both the sum and differences of the two numbers are divisible by the HCF.
·          If numbers N1, N2, N3, N4 etc. give an equal remainder when divided by the same number P, then P is a factor of (N1 - N2), (N2 - N3), (N3 - N4)...
·          If L is the LCM of N1, N2, N3, N4.. all the multiples of L are divisible by these numbers.
·          If a number P always leaves a remainder R when divided by the numbers N1, N2, N3, N4 etc., then P = LCM (or a multiple of LCM) of N1, N2, N3, N4..  + R.

SOLVED PROBLEMS ON LCM

Find the highest four-digit number that is divisible by each of the numbers 24, 36, 45 and 60.
Answer: 24 = 23
× 3, 36 = 22 × 32, 45 = 32 × 5 and 60 = 23 × 32 × 5.
Hence, the LCM of 24, 36, 45 and 60 = 23
× 32 × 5 = 360.
The highest four-digit number is 9999. 9999 when divided by 360 gives the remainder 279. Hence, the number (9999 – 279 = 3720) will be divisible by 360.

Hence the highest four-digit number divisible by 24, 36, 45 and 60 = 3720.

Find the highest number less than 1800 that is divisible by each of the numbers 2, 3, 4, 5, 6 and 7.

Answer: The LCM of 2, 3, 4, 5, 6 and 7 is 420. Hence 420, and every multiple of 420, is divisible by each of these numbers. Hence, the number 420, 840, 1260, and 1680 are all divisible by each of these numbers. We can see that 1680 is the highest number less than 1800 which is multiple of 420.
Hence, the highest number divisible by each one of 2, 3, 4, 5, 6 and 7, and less than 1800 is 1680.

Find the lowest number which gives a remainder of 5 when divided by any of the numbers 6, 7, and 8.

Answer: The LCM of 6, 7 and 8 is 168. Hence, 168 is divisible by 6, 7 and 8. Therefore, 168 + 5 = 173 will give a remainder of 5 when divided by these numbers.

What is the smallest number which when divided by 9, 18, 24 leaves a remainder of 5, 14 and 20 respectively?

Answer: The common difference between the divisor and the remainder is 4 (9 - 5 = 4, 18 - 14 = 4, 24 - 20 = 4). Now the LCM of 9, 18, and 24 is 72.

Now 72 - 4 = 72 - 9 + 5 = 72 - 18 + 14 = 72 - 24 + 20. Therefore, if we subtract 4 from 72, the resulting number will give remainders of 5, 14, and 20 with 9, 18, and 24.

Hence, the number = 72 - 4 = 68.

A number when divided by 3, 4, 5, and 6 always leaves a remainder of 2, but leaves no remainder when divided by 7. What is the lowest such number possible?

Answer: the LCM of 3, 4, 5 and 6 is 60. Therefore, the number is of the form 60k + 2, i.e. 62, 122, 182, 242 etc. We can see that 182 is divisible by 7. Therefore, the lowest such number possible = 182.

I shall have to end here and leave the rest of it for my CBT Club students. I shall cover some problems based on this in the CBT Club this week.

 

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Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by vikas sharma - Tuesday, 17 February 2009, 12:26 PM
 

hi TG sir,

when we have to find lcm/HCF in language problem.Like above in soldier prob.How do we conclude its case of HCF/LCM ,can u pls brief it.

Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by ankita singh - Thursday, 26 February 2009, 01:35 PM
  Hi!

I think there is some problem with the solution of the question that involves finding out the highest four digit number that is divisible by 24,36,45 and60.

The answer should be  9720 instead of 3720.

Respond ASAP!!!

Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by Total Gadha - Friday, 27 February 2009, 10:46 AM
  Hi Ankita,

Yes you are correct. 3720 is a typo. 9720 is the correct answer.

Total Gadha
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by vikas sharma - Monday, 2 March 2009, 01:16 PM
 

hi TG sir,

when do we have to find lcm/HCF in language problem.Like above in soldier prob.How do we conclude its case of HCF/LCM ,can u pls brief it

Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by arunika bedi - Friday, 15 May 2009, 08:33 PM
  hi tg sir...
i think the ans 2 the ques
find hcf of 36x3y2 and 24x4y.
the ans shud be 12
x3y instead of 12x2y.
plzz make a checck to it.n make it correct..
thanku
yours sincerely,
arunika.
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by sujay singh - Friday, 22 May 2009, 03:47 PM
  Hi vikas

This is my first ever reply to an post on TG..

When The quesion involves making of Least such groups etc as in case of soldier problem , U need to apply HCF because that will provide u the lowest nos when used as a divisor.

LCM ke liye bhi aisa he kuch dimak lagao ....

waise there are not many diff. type that come under these two category ... practise from net questions are enough for LCM , HCF qns..

Was srry to see that TG didnt reply to ur qn , inspite of u wriing it in Bold the second time ..


regards
Sujay
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by Total Gadha - Monday, 25 May 2009, 04:26 AM
 

Hi Sujay,

Thanks for replying to the query. smile

I nearly always reply to a query sooner or later. But somebody writing in bold and caps feels offensive to me.

 

Total Gadha

Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by Deepika Khandelwal - Tuesday, 26 May 2009, 05:40 PM
  Hi TG,
this really very useful article....
please give some more questions on HCF and LCM to practice..

Thanks
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by saurabh yadav - Friday, 26 June 2009, 05:16 PM
  What is the smallest number which when divided by 9, 18, 24 leaves a remainder of 5, 14 and 20 respectively?

Answer: The common difference between the divisor and the remainder is 4 (9 - 5 = 4, 18 - 14 = 4, 24 - 20 = 4). Now the LCM of 9, 18, and 24 is 72.

Now 72 - 4 = 72 - 9 + 5 = 72 - 18 + 14 = 72 - 24 + 20. Therefore, if we subtract 4 from 72, the resulting number will give remainders of 5, 14, and 20 with 9, 18, and 24.

Hence, the number = 72 - 4 = 68.


logic is not clear , plz elaborate it.

thnx.

Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by Antonio Banderas - Friday, 26 June 2009, 10:31 PM
 

my method to this is as follows

let d number be n

so we have n-5 divisible by 9

                 n-14 divisible by 18

        and    n-20 divisible by 24

thus we can write 9*a+5=18*b+14 where we see least number satisfying the equation is a=3 and b=1..

thus number satisfying the 1st 2 constraints will be of the form

        18k+32

(taking L.C.M of 9,18 and getting the value of 9*a+5 or 18*b+14 putting a=3or b=1)

similarly we compare the 3rd condition with what we get

thus we have 18k+32=24c+20 ,and we get k=2 c=2

thus the final form of the number should be

 72k+68 (L.C.M of 18,24 and putting k=2 in 18k+32)

so the minimum number is 68 for k=0.

This is a long process, a better approach when the difference between remainder and divisor is same will be

   (L.C.M of the divisors)-(the difference)

here L.C.M is 72 and difference is 4 so directly we get 72-4=68.

Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by William Wallace - Monday, 20 July 2009, 08:45 PM
  Hey friends , what would be the LCM of 3333....3 (written 70times) and 2222....2(written 20 times)?

Any clues?
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by vikram solanki - Tuesday, 21 July 2009, 07:16 PM
  is its answer 140 * 6 = 840..
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by abhishek srivastava - Sunday, 26 July 2009, 01:06 PM
  referring to the property i.e [i]·          If numbers N1, N2, N3, N4 etc. give remainders R1, R2, R3, R4, respectively, when divided by the same number P, then P is the HCF of (N1 - R1), (N2 - R2), (N3 - R3), (N4 - R4) etc.[/i].i have one doubt .



if n1,n2 ,n3.... no. when divided  by p  give remainder r1,r2,r3..... then it can be writtern as
        
         n1 = px + r1
         n2 = py + r2
         
and so on...
      
         thus ,
               
          n1-r1 =px
          n2-r2 =py

and so on...

         therefore hcf of (n1-r1),(n2-r2)....is not necessarily p since it is not necessary that x,y is coprime (unless p is the highest no ,which when divides n1,n2......gives remainder r1,r2.)...
              
         i hope you will clear my doubt.
 

Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by Ching Lee - Tuesday, 28 July 2009, 03:09 PM
 

A typing mistake...

Find the highest four-digit number that is divisible by each of the numbers 24, 36, 45 and 60.
The answer should be 9999 - 279 = 9720, and not 3720.

Regards.

Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by Satish Kumar - Tuesday, 28 July 2009, 09:49 PM
  Good and concise information...........
But there is some typo error exist
For Example.
The highest four-digit number is 9999. 9999 when divided by 360 gives the remainder 279.
 Hence, the number (9999 - 279 = 9720) will be divisible by 360.

Hence the highest four-digit number divisible by 24, 36, 45 and 60 = 9720 not 3720.

Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by Veena Binya - Saturday, 22 August 2009, 08:48 AM
  Hi TG

Under the head 'PROPERTIES OF HCF AND LCM', there is this property that you have mentioned...

'If numbers N1, N2, N3, N4 etc. give remainders R1, R2, R3, R4, respectively, when divided by the same number P, then P is the HCF of (N1 - R1), (N2 - R2), (N3 - R3), (N4 - R4) etc.'

I considered the case where

N1 N2 N3 N4 are
4 8 12 16

and P is 2

Therefore R1 R2 R3 R4 would be
0 0 0 0

Now (N1 - R1) (N2 - R2) (N3 - R3) (N4 - R4) would be
4 8 12 16

The HCF of these turns out to be 4, and not 2. Therefore P is not the HCF of (N1 - R1), (N2 - R2), (N3 - R3) and (N4 - R4) ?
It seems like it is going to be only a CF(common factor)

Kindly correct me if I am wrong.
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by nitish goel - Sunday, 27 September 2009, 02:04 PM
 

hi tg,

i think the answer to this question on lcm given as below shud be 5.

Find the lowest number which gives a remainder of 5 when divided by any of the numbers 6, 7, and 8.
plz respond

Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by sammy badyal - Sunday, 27 September 2009, 05:26 PM
  agreed
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by sonal tanna - Wednesday, 14 October 2009, 06:53 PM
 

specific to soldier problem, the answer cannot be LCM because the soldiers in a group cannot be greater than two numbers--(LCM is greater than or equal to the largest number).

Hence a commonsense can help you find out. Just remember HCF<n1,n2<LCM.

in some cases, HCF=least(n1,n2)

LCM=greatest(n1,n2)

Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by Nikhil Jain - Tuesday, 20 October 2009, 11:05 PM
  hi solanki,

how is it ?
Can u explain how u got 140?
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by krd sh - Monday, 2 November 2009, 11:45 AM
 

The answer to the question put by Nitish Goesl  is 173. Following is the method to do it:

Take the L.C.M of 6, 7 and 8 that will come out to be 168

Lowest number = 168m + 5

we have to put the lowest possible value in m

I have put 1 which is the lowest possible value.

On putting 1 we get 173. Check by dividing 173 by 6, 7 and 8 respectively.

Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by karan khetan - Friday, 6 November 2009, 06:06 PM
  u have to select even numbers from 1 to 25 and odd numbers from 26 to 200 and if u take a product of them how many zeros will be there in the product...can someone answer this question...???
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by Anurag singhal - Monday, 9 November 2009, 02:10 PM
 

Hi frineds ,

I am not able to understand the logic behind (PFB) question: can any one elaborate on this . 

What is the smallest number which when divided by 9, 18, 24 leaves a remainder of 5, 14 and 20 respectively?

Answer: The common difference between the divisor and the remainder is 4 (9 - 5 = 4, 18 - 14 = 4, 24 - 20 = 4). Now the LCM of 9, 18, and 24 is 72.

Now 72 - 4 = 72 - 9 + 5 = 72 - 18 + 14 = 72 - 24 + 20. Therefore, if we subtract 4 from 72, the resulting number will give remainders of 5, 14, and 20 with 9, 18, and 24.

Hence, the number = 72 - 4 = 68.

 

Regards,

Anurag

Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by Total Gadha - Monday, 9 November 2009, 08:06 PM
  Hi Anurag,

Just understand it in a simpler manner- The number is giving negative remainder of -4 with the divisors. So the number = LCM(divisors) + remainder

Total Gadha
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by Anurag singhal - Monday, 9 November 2009, 10:41 PM
  Hi TG ,

Got it by your reply.

In btw: I am turning pages after pages on this site only for last 5 days , since i come to know about this  And stuff given here is tempting me to turn more

Gr8 compilation of smart questions and formula's smile
 
Thanks and regards,
Anurag
plz solve this...
by reema kharbanda - Saturday, 5 December 2009, 05:30 PM
  lcm of 4.5, .009 and .18?
Re: plz solve this...
by Anjan rajan - Wednesday, 16 December 2009, 03:47 PM
  45/10,9/1000,18/100

LCM= LCM(Num)/HCF(Den)
      = 90/10 = 9
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by prasanth warrier - Monday, 18 January 2010, 10:44 AM
  iam prashant..iam a B.tech student studying in 5th sem mumbai univ.
i found dis site the only reliable source..iam of the same thought that all classes r waste if we have such source to study..
the thing which i found abt classes are that dey dont make us think rather we think in de way they thought..good for them but hate such ..but now iam studying alone..i need to make a clear schedule of how to go abt it..i wish to complete my studies by august end...pls help me make a timetable according to ur uploaded files..i will surely stick to it pls help me out tg..
eagerly waitin for ur reply
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by manendra khurana - Saturday, 13 February 2010, 02:59 PM
 

hi ! TG sir

 

Re: plz solve this...
by manendra khurana - Saturday, 13 February 2010, 03:20 PM
  hi reema , ur ans is 4.5
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by Rajratan Singh - Monday, 1 March 2010, 12:38 AM
 

Thanks man!!!

this article is just awesome.

Re: plz solve this...
by aditya khorana - Tuesday, 2 March 2010, 03:24 PM
 

@ manendra

i feel the answer should be 90/10 i.e. 9

the LCM of numerator would be 90

and the HCF 10

so LCM of the fraction becomes 9.

Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by aditya khorana - Tuesday, 2 March 2010, 04:26 PM
 

Hi TG sir,

Please explain how the answer is 140 to the question asked by William?

Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by bhavana Pandey - Tuesday, 2 March 2010, 08:24 PM
  Hello, TG sir.
i have a query.. When we have to find the factors of a number which are coprime to each other and find such pairs of number, is there any easy way to do this?? For example- in questions in which the HCF and the sum of two numbers is given, n we need to find the number of pairs possible..
Sometimes it becomes tedious to figure out which factors are coprime to each other.
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by Rahul Bhat - Friday, 5 March 2010, 05:36 PM
 

Yes Nitish u r rite...

Why the need for LCM in this case then?

Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by Total Gadha - Sunday, 7 March 2010, 12:35 AM
  Hi Bhavna,

See the video: http://totalgadha.com/mod/resource/view.php?id=1147
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by xlr pmir - Wednesday, 28 April 2010, 08:53 PM
  i find the article copied from a book called arihant, or arihant has taken it from tg sir. but every concept and even the questions in the book and article are exactly common.tongueout
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by chetan chandrashekar - Saturday, 21 August 2010, 12:32 PM
  Hi xlr pmir,

Its ok even if things are copied as this can be the one stop for the best concepts spread across.Lets always learn to appreciate the effort,not all will know about the book, arihant and not all of us will buy it either smile
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by srinivas hosur - Sunday, 17 April 2011, 08:18 PM
  to get a zero in the units place,u need to multiply any no. with 10 or 5*2.
no of 2's in the interval (1,25)=21(only even)
no of 5's in the interval (6,200)=21(only odd)
therefore no. of 0's=21............

Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by srinivas hosur - Monday, 18 April 2011, 05:07 PM
  hi TG sir,
plz tell me how to solve this problem
1.find the least number which when divided by 4,5,11,13 leave a remainder 1,2,4,11 resp.(ans:37)
if the remainder is same in all cases ,i know the shortcut but i am unable to solve the above problem.
please help me out with this problem.
thank you.
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by TG Team - Tuesday, 19 April 2011, 11:04 AM
 
Hi Srinivas smile

You can combine the different remainders using Chinese Remainder Theorem. But best way is to write all such natural numbers which give the desired remainder with largest divisor.

In this case N = 11 mod13, so N can be any one of these: 11, 24, 37, 50, .....
Checking these numbers with 11 as divisor first, we get that 37 = 4 mod11 as desired.

So N = 37 mod(11*13), that means N can be anyone of these: 37, 180, 323, ...
Checking these numbers with 5 and 4, we find that 37 is the smallest number which satisfies all the given conditions, hence answer.

By the way, if you have got options for the answer, then there is no need to do all this. Just check all the options individually and they must satisfy all the given remainder conditions.

Hope this helps. smile

Kamal Lohia   

Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by ankit khandelwal - Wednesday, 4 May 2011, 10:23 PM
  Sir,
how to calculate the LCM of decimal numbers?

Ankit Khandelwal
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by TG Team - Thursday, 5 May 2011, 11:38 AM
 
Hi Ankit smile

First we need to write the decimal numbers in fraction form(lowest form i.e. numerator and denominator are co-prime), then we can find that
LCM(a/b, c/d, e/f) = LCM(a, c, e)/ HCF(b, d, f), AND
HCF(a/b, c/d, e/f) = HCF(a, c, e)/ LCM(b, d, f).

e.g. LCM(0.5, 0.4, 0.7) = LCM(1/2, 2/5, 7/10) = LCM(1, 2, 7)/ HCF(2, 5, 10) = 14
and HCF(0.5, 0.4, 0.7) = HCF(1/2, 2/5, 7/10) = HCF(1, 2, 7)/ LCM(2, 5, 10) = 1/10 = 0.1 smile

Kamal Lohia
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by srinivas hosur - Saturday, 7 May 2011, 11:41 PM
  hi kamal
thanks for the soln.
i have one more query.....
1)x+y=n,(x,y) are pair of co primes,how many such pairs r possible....
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by destiny unruled - Monday, 9 May 2011, 11:58 PM
  @ Srinivas

First I have to explain Euler's number, then I'll explain the question.

Euler's Number

Euler's number, φ(n), of a a number n is the number of positive integers less than or equal to n that are coprime to n.

If n = (p1)^a*(p2)^b*(p3)^c*......, where p1, p2, p3, .... are prime numbers then
φ(n) = n(1 - 1/p1)(1 - 1/p2)(1 - 1/p3).....

For ex:-
147 = 3*7^2
φ(147) = 147(1 - 1/3)(1 - 1/7) = 84
Hence, there are 84 numbers less than 147 and co-prime to 147

Now the question is:
x + y = n, where x and y are co-prime. We have to find the number of such pairs.

We know that if 'x' is co-prime to n, then (n - x) will also be co-prime to n.

So, we just have to find the number of numbers less than n and co-prime to n.

But we know that number of numbers less than n and coprime to n is given by Euler's number.

So, required number of numbers = φ(n)

But this will give us the number of ordered pairs of (x, y).

If just number of pairs are asked then, (p, q) and (q, p) both are same. So, answer will be φ(n)/2.

I hope it makes sense
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by satya rachakonda - Thursday, 11 August 2011, 09:39 PM
  L.C.M of two numbers is 666....140 times

procedure:
L.C.M: l.c.m of 2 and 3 is 6.since 2 and 3 are recurring numbers in given numbers.so we have to take lcm of two numbers.lcm is 6.so in lcm 6 will repeat.we have to find how many times "6" will appear.now find l.c.m of powers of given numbers(so as to find the no. of times 6 will appear)l.c.m of 20 and 70 is 140.so the answer is (l.c.m) is
6666.....140 times.
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by neha aggarwal - Thursday, 19 July 2012, 08:28 PM
  Hi Sir smile,

Please help me with this:

How many ordered triples (a, b, c) are there, such that lcm(a, b) = 1000, lcm(b, c) = 2000, lcm(c, a) = 2000?

Thanks,
Neha
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by TG Team - Friday, 20 July 2012, 05:02 AM
 

Hi Neha smile

It's clear that a, b, c will contain only powers of 2 and 5. So just assign them some variable powers and give these variables some values according to given conditions.

Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by rohith kotagiri - Thursday, 27 September 2012, 09:19 AM
  Dear TG,

How to solve the following?
Find the HCF of [(2^100)-1 , (2^120-1)]
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by Arun Venkataramani - Saturday, 27 October 2012, 10:18 AM
  Hi,

I think the answer is 5.

2^100= (2^10)^10= ends in 76 .
so 76 -1 =75.
similarly 2^120 also ends in 76 and -1 becomes 75.
Both the numbers are multiples of 5, hence HCF should be 5.

property to be known= (2^10)^even ends in 76.
Re: Highest Commom Factor (HCF) and Least Common Multiple (LCM)
by abbhinav addanki - Thursday, 22 January 2015, 12:25 PM
  thank you mr tg sir your post is very useful.