Question

Prove that HCF x LCM = Product of two numbers.

Answer 1

can be proved by using examples- as
let the two no. be X & Y and,
X = a x b
Y = c x d
LCM of X&Y will be equal to = XY = a.b.c.d
HCF of " " " " " " " = 1
Now, ATQ,
HCF x LCM = a.b.c.d ............1
& X x Y = a.b.c.d ................2
from 1 & 2 we get,
HCF x LCM = X x Y

Answer 2

Given as :

HCF and LCM

And, Two numbers

To Prove :

Products of two numbers = The product of HCF and LCM

Solution :

Let, x and y be two numbers ,

Let HCF of numbers = H

Let LCM of numbers = L

we dived x and y by H , so the respective quotient be m and n

And m , n are prime to each other .

So, x = m H  ,  y = n H

∴  LCM = L = m n H

Now, Products of numbers

  x × y = m H × n H

          = H ( m × n  × H)

          = H × L

∴   x × y = H × L

Or, Products of number x , y = Product of HCF and LCM   proved

EXAMPLE :

let two numbers = 12 and 16

So, LCM of 12 ,1 6 = 3 × 4 × 4

i.e   LCM of 12 ,1 6 = 48

And

HCF of 12 , 16 is

Factor of 12 = 2 × 2 × 3

Factor of 16 = 2 × 2 × 2 × 2

i.e, HCF of 12 , 16 =  2 × 2 = 4

Now,

Products of numbers = 12 × 16 = 192

Products of HCF and LCM = 48 × 4 = 192

Thus, It is proved that - Products of numbers = Products of HCF and LCM

Hence,  Products of numbers = Products of HCF and LCM  . Proved