### 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 QUESTIONS 1
 HCF AND LCM QUESTIONS 2

 HINTS AND SOLUTIONS 1

 HINTS AND SOLUTIONS 2

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