Question

If two positive integers p and q are written as p=a²b³ and q=a³b; a and 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

I hope this helps you.

: )