H.C.F and L.C.M

Facts And Formulae: 

Highest Common Factor:(H.C.F) or Greatest Common Meaure(G.C.M) : 
The H.C.F  of two or more than  two numbers is the greatest 
number that divides each of them exactly.

There are two methods :

i.Factorization method:  Express each one of the given numbers as 
the product of prime factors. The product  of least powers of common
prime factors gives HCF.

Example :  Find HCF of 26 * 32*5*74 ,   22 *35*52 * 76 , 
           2*52 *72
Solution:  The prime numbers given common numbers are  2,5,7
           Therefore HCF  is  22 * 5 *72 .

ii.Division Method : Divide  the larger number by smaller one. Now 
divide the divisor by remainder. Repeat the process of dividing
preceding number last obtained till zero is obtained as number. The
last divisor  is HCF.

Example:   Find HCF of 513, 1134, 1215
     
Solution:  

     1134) 1215(1
           1134
         ----------
             81)1134(14
                 81
              -----------
                  324
                  324
               -----------
                   0
              -----------
    HCF of this two numbers is 81.

       81)513(6
          486
           --------
           27)81(3
              81
             -----         
               0                 
              ---

HCF  of  81 and 513 is 27.


Least common multiple[LCM] : The least number which is 
divisible by each one of given numbers is LCM.

There are two methods for this:

i.Factorization method  : Resolve each one into product of prime 
factors. Then LCM is  product of highest powers of all factors.

ii.Common  division method.
                                                    
Problems:

1.The  HCF of 2   numbers  is 11 and LCM is  693.If one of numbers 
is 77.find other.

Sol:  Other number  = 11 *  693/77=99.

2.Find largest number of 4 digits  divisible by  12,15,18,27

Sol: The largest number is 9999.
LCM of 12,15,18,27  is 540.
on dividing 9999 by  540 we get 279 as remainder.
Therefore 
number =9999 – 279  =9720.

3.Find least number which when divided by 20,25,35,40 leaves 
remainders 14,19,29,34.

Sol:        20–14=6
       25-19=6
       35-29=6
       40-34=6
Therefore number =LCM of (20,25,35,40) - 6=1394

4.252  can be expressed as prime as :
                     2      252
                     2      126
		     3      63
		     3      21
	                     7
prime factor is 2 *2 * 3 * 3  *7

5.1095/1168  when expressed in simple form is   
    1095)1168(1
         1095
         ------
           73)1095(15
               73
            ---------
               365
               365
              ---------
                0
             ----------
So, HCF is 73
Therefore 
1095/1168 = 1095/73/1168/73= 15/16

6.GCD of 1.08,0.36,0.9 is
 
Sol: 
HCF of 108,36,90         36)90(2
           72
           ---- 
	   18)36(2
              36
             ----
              0
            ----
HCF is 18.
HCF of 18 and 108 is 18
  18)108(6
     108
    -------
       0
    --------
Therefore  HCF  =0.18

7.Three numbers are in ratio 1:2:3 and HCF is 12.Find numbers.

Sol: 
Let the numbers be x.
Three numbers are x,2x,3x
Therefore 
HCF is            2x)3x(1
                     2x
		     -----
                      x)2x(2
                        2x
                      --------
		        0
		   -------------	

HCF is x so, x is 12
Therefore numbers are 12,24,36.
                                                    
8.The  sum of two numbers is 216 and HCF is 27.

Sol:   Let numbers are 27a  + 27 b =216
                  a + b =216/27=8
Co-primes of 8 are (1,7)  and (3,5)
numbers=(27  * 1 ), (27  *  7)
       =27,89

9.LCM of two numbers is  48..The numbers are in ratio 2:3. The sum
of numbers is

Sol:  
Let the number be x.
Numbers are 2x,3x
LCM of  2x,3x is 6x
Therefore 
6x=48
x=8.
Numbers are 16 and 24
Sum=16 +24=40.

10.HCF and LCM of two numbers are  84 and 21.If ratio of two numbers
is 1:4.Then largest of two numbers is  

Sol:  
Let the numbers be x,4x
Then x *  4x  =  84  *  21
  x2 =84  * 21 /4
           x  = 21
 Largest number is 4 * 21.

11.HCF of two numbers is 23,and other factors of LCM  are 13,14.
Largest number  is

Sol:  
 23  * 14 is Largest number.

12.The maximum number of students among   them 1001 pens and 910 
pencils can be distributed  in such a way that each student  gets 
same number of pens and pencils is ?

Sol: 
HCF of 1001 and 910
   910)1001(1
        910
       ------------
              91)910(10
                 910
                   --------
                      0
                  ---------
       Therefore  HCF=91

13.The least number which should be added to 2497 so that sum is
divisible by 5,6,4,3  ?

Sol:  LCM   of  5,6,4,3   is 60.
On dividing  2497  by 60   we get 37 as remainder.
Therefore number to added is 60 – 37  =23.
Answer is 23.

14.The least number which is a perfect square and is  divisible by 
each of numbers 16,20,24 is ?

Sol:  LCM of 16,20,24   is 240.
2 * 2*2*2*3*5=240
To make it a perfect square multiply by 3 * 5
Therefore  240 * 3 * 5=3600
Answer is 3600.