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HCF AND LCM math capsule

Formulas you need to remember :


1. when you want to calculate the least number which when gets divided by numbers namely x,yand z and leaves remainders as a,b and c respectively, then the number can be calculated as ,

                      number = [LCM of (x,y,z)-p], where p=(x-a)=(y-b)=(z-c)

2.when you want to calculate the least number which when divided by x,y and z leaves the same remainder in each case , then the number will be,
 
                      number =[LCM of (x,y,z)+R], where R is the remainder left in each case



3.   LCM(a,b) * HCF(a,b) = product of the two numbers that is( a*b)
     =>HCF = (a*b)/LCM
     =>LCM=(a*b)/HCF

4.  LCM and HCF of fractions :
   
          LCM of fractions = LCM of numerators/ HCF of denominators

e.g. LCM of  (5/7,6/5) = LCM(5,6)/HCF(7,5)

           HCF of fractions = HCF of numerators/ LCM of denominators

e.g. HCF of (3/7,4/5) = HCF (3,4)/LCM(7,5)


5.  when you want to calculate the greatest number that will divide x,y and z leaving the remainders as  a,b and c , respectively then the number will be the ,  HCF ((x-a),(y-b),(z-c))

6. when you want to calculate the greatest number that will divide numbers x,y and z leaving the same remainder in all the cases , the number will be equal to the , HCF (|x-y|,|y-z|,|z-x|)
    ,where |x| is the modulus operation .

HOW TO CALCULATE HCF AND LCM : 



LCM 


There are two method for calculating the lcm of two numbers _ (1) Prime Factorisation Method (2) LEAST COMMON MULTIPLE Factorisation Method

    → Prime factorisation for calculating lcm
In this method we write down the prime factors of the numbers , then the LCM is the product of the highest powers of all the factors.

e.g Q. Calculate the LCM of 25, 30, 40.
 
         so the factors are ,
            25 = 5*5=(5²)  , 30= 5*2*3, 40=5*2*2*2=5*(2³), where bª means b to the power of a
           LCM of 25,30,40 = (5²)*(2³)*(3¹)= 600
 
    → Least Common Multiple of more than two numbers by Factorisation
in this method you divide all the numbers or as many as possible by such a prime common divisors as may be contained in them and then you multiply the divisors together and the final quotients.

e.g. Q. Calculate the LCM of 15 , 14 and 20.

       5 | 15,14,20
       2 | 3,14,4
       7 | 3,7,2
       3 |3,1,2
       2 |1,1,2
          |1,1,1

Now , LCM (15,14,20)=  5*2*7*3*2=420

HCF  


HCF can be calculated by using two methods _ (1) Common Factor Method (2) Highest Common Factor

   → Common Factor : a common factor of two or more than two numbers which divides each of them is calculated

HCF of (5,3,2)= 1
HCF of (3,6,9)=3
HCF of (25,30,40)= 5

  →Division method
 HCF of 1445, 1190 =

                1190)1445(1
                         1190
                          255)1190(4
                                  1020
                                   170)225(1
                                          170
                                           85)170(2
                                                170
                                                   0

so , the HCF is 85 .


QUESTIONS TO PRACTICE :

HCF AND LCM 1
HCF AND LCM QUESTIONS 1
HCF AND LCM 2
HCF AND LCM QUESTIONS 2

HCF AND LCM 3
HINTS AND SOLUTIONS 1

HCF AND LCM 4
HINTS AND SOLUTIONS 2



   
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