HCF & LCM

Highest Common Factor &Least Common Multiple 

1. HCF:

     If the numbers N1 and N2 are exactly divisible by x then x is said to be the common factor of N1 &N2. The highest of all the common factor of N1 &N2 is called as HCF Of two numbers.
Ex: Find out the HCF of 24&36
  Factors of 36= 1,2,3,4,6,9,12,18and 36
   Factors of 24=1,2,3,4,6,8,12and 24
Common factor =1,2,3,4,6,12.
Highest of them is 12. So that,
  HCF = 12.

2. LCM :

   Let N1 and N2 be two natural numbers. The smallest number n that is exactly divisible by N1 &N2 is called as least common multiple of two numbers.
Ex: Find the LCM of 16 and 12
     Multiples of 16: 16,32,48,64,80,96,112,128,144,160,..
 Multiples of 12: 12,24,36,48,60,72,84,96,..
Common Multiples: 48,96,...
LCM: 48.

3. Some Formulae :

a. HCF *LCM = 1NO * 2No.
b.HCF of co prime numbers= 1.
c. LCM of co prime numbers = Their product.

4. Applications of HCF and LCM

i. Find the greatest number that will exactly divide a,b and c= HCF(a,b,c).

ii. Find the greatest number that will divide a,b and c leaving remainder of x,y and z respectively = HCF (a-x,b-y,c-z).

iii.  Find the greatest number which when divides a,b and c will leave the same remainder in each case = HCF (a-b,b-c,c-a).

iv.  Find the least number which is exactly divisible by a,b and c= LCM(a,b,c).

v.  Find the least number which when divided by a,b and c leaves the same remainder r in each case = LCM (a,b,c)+r.

vi. Find the least number which when divided by a,b and c leaves the remainder x y and z respectively = Check if a-x= b-y=C-z=K. If this is the case then LCM (a,b,c)-K.