Question

If two positive integers p and q are written as p=a^2b^3 and q=a^3b ; a , b are prime numbers then verifylcm(p,q)*hcf(p,q)=pq

Answer 1

p = a²b³

q = a³b


HCF ( p , q ) = a³b³

LCM ( p , q ) = a²b


To verify :-

LCM ( p , q ) × HCF ( p , q ) = pq


LHS :-

LCM ( p , q ) × HCF ( p , q )

a²b × a³b³


$= {a}^{5} {b}^{4}$

RHS :-

pq

a²b³ × a³b

$= {a}^{5} {b}^{4}$


LHS = RHS

Answer 2

HCF :- the highest power

LCM :- the lowest power.

So,

HCF ( p , q ) = a³b³

LCM ( p , q ) = a² b

To verify :-

LCM ( p , q ) × HCF ( p , q ) = pq

Taking LHS

LCM ( p , q ) × HCF ( p , q )

a² b × a³ b³

$= {a}^{5} {b}^{4}$


Taking RHS

pq


a² b³ × a³ b

$= {a}^{5} {b}^{4}$

Since , LHS = RHS

hence verified .