HCF and LCM

HCF means highest common factor or greatest common divisor.

LCM means the least common multiple.

HCF of two or more than two number can be find by prime factorization method.

Whereas the LCM is the multiplication of common factors as well as the which are not 
common.  

     1) Find HCF of 12, 18 and 27

Solution:- by prime factorization method
12=3*2*2
18=3*3*2
27=3*3*3
HCF= factors which are common in all the three number
HCF=3

But the method is not applicable in case of having large numbers.
So there is a method known as ‘long division method’ to find the HCF of large numbers.

 Method:-

1: find the smallest and biggest number from the given number.

2: consider the smallest number as the divisor and the biggest number as dividend.

3: divide them and get the remainder.

4: check whether the remainder is zero or not.

Case 1: if the remainder is zero then stop the procedure. The divisor of the step having remainder zero is the required HCF.

Case 2: if the remainder is not zero then consider this remainder as the divisor and the divisor of the previous step as the dividend and follow the step 3 & 4.

     2)Find the HCF of 3442 and 4267

Solution:- consider the smallest number as the divisor and the large number as the dividend.

3442) 4268 (1
         -3442
         ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞
             826) 3442 (4
                    -3304
                     ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞ 
                      138) 826 (5
                             -690
                          ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞
                               136 ) 138 (1
                                       -136
                                            ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞   ͞
                                           2 ) 136 (68
                                                -136
                                                ͞   ͞   ͞   ͞   ͞
                                                   000

      Hence the HCF=2 

      

     3) Find the LCM of 25 and 60.

Solution:- by prim factorization method

25=5*5
60=5*4*3
LCM=5*5*4*3
LCM=300

But the method is limited. If we have given big or more than 2 number then it would be very difficult to find the LCM.

There is one method which can be used to find the LCM of two or more (big) numbers.

Method:-

1: write down the given numbers in a tabular form.

2: either select a prim number which divides at least two of them or a composite 
 number which divides all the numbers.

3: if you have selected a prime number which divides two of them then simply do the division and write the quotient in the next row and if the numbers which are not divisible by the number ‘which has been selected as the divisor’ then write those numbers as it is.

4: follow the step 2 & 3 again and again till you reach to the step where you don’t find any divisor.

5: now the LCM=product of numbers in the first column and the last row

     

     4)    Find the LCM of 20, 25, 30 & 40

Solution:- write down the given numbers in a tabular form.
_____|__20__|__25__|__30__|__40__|
__5__|__4___|__5___|__6___|__8___|
__2__|__2___|__5___|__3___|__4___|
__2__|__1___|__5___|__3___|__2___|

LCM=5*2*2*1*5*3*2=600

     5)    A farmer has 20 apples plants, 35 orange plants and 75 strawberry plants. He wanted to arrange them in rows such that each row has the same number of trees and all are of same type. So find the minimum number of rows that can be formed?

Solution:-  here we first find the number of plants can be planted in a row
That means we will have to find the HCF of smallest and the biggest number from 20,35 & 75

So we find HCF of 20 & 75 by long division method.

20)75(3
    -60
   ͞   ͞   ͞   ͞
      15) 20 (1
           -15
         ͞   ͞   ͞   ͞   ͞
             5)15 (3
               -15
                ͞   ͞   ͞
                00

Hence the HCF=5

=> 5 plants can be planted in one row

Total number of rows=total number of plants / number of plants in one row
Total number of rows=(20+35+75)/5=26 

6)     There are 5 bells which tolls after each 10 sec, 12 sec, 20sec, 25sec and 40 sec. At what time will they all toll together?

Solution:- here we have to find the LCM

_____|__10__|__12__|__20__|__25__|__40__|
__2__|__5___|__6___|__10__|__25__|__20__|
__2__|__5___|__3___|__5___|__25__|__10__|

LCM=2*2*5*3*5*25*10=75000 sec 

So far we have seen how to find the LCM & HCF of integer number. Now we 

will see how to find the LCM & HCF of fractions.

HCF of fraction= HCF of numbers in numerator / LCM of numbers in denominator

LCM of fraction= LCM of numbers in numerator / HCF of numbers in denominator

1)    Find the LCM of 0.25 and 1.55

Solution:- we can write 0.25 and 1.55 as follows.

25/100=1/4 and 155/100=31/20

LCM of fraction= LCM of numbers in numerator / HCF of numbers in denominator

LCM of fraction= LCM ( 1,31)/ HCF (4, 20)=31/4

2)    Find the HCF of 0.45 and 2.15

Solution:- 0.45=45/100=9/20 and 2.15=215/100=43/20

HCF of fraction= HCF (9,43) / LCM (20,20)=1/20

Note:- it is necessary to write down the given fractions into the simplest form.