Question

If two positive integers p and q are written as
2 3 3 p a b and q a b; = =
a, b are prime
numbers, then verify:
LCM (p, q) × HCF (p, q) = pq

Answer 1

Hi ,

p = a²b³

q = a³b

HCF ( p,q ) = a²b

 [ ∵Product of the smallest power of each

      common prime factors in the numbers ]

LCM ( p , q ) = a³b³

[ ∵ Product of the greatest power of each 

   prime factors , in the numbers ]

Now ,

HCF ( p , q ) × LCM ( p , q ) = a²b × a³b³

                                             = a∧5b∧4 --------( 1 )

[∵ a∧m × b∧n = a∧m+n ]

pq = a²b³ × a³b

     = a∧5 b∧4 ---------------( 2 ) 


from ( 1 ) and ( 2 ) , we conclude 

HCF ( p , q ) × LCM ( p ,q ) = pq

Hope This Helps :)