Question

if two positive integers p and q are written as p= a^2 b^3 and q= a^3 b; a, b are prime numbers, then verify: LCM (p, q) x 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.

: )

Answer 2

Given that,
p = a² b³
q = a³ b
We have to prove that,
L.C.M (p,q) x H.C.F (p,q) = p q
Proof:
Since,
p = a² b³
p = a.a.b.b.b
q = a³ b
q = a.a.a.b
H.C.F (p,q) = a.a.b
H.C.F (p,q) = a² b
L.C.M (p,q) = a.a.a.b.b.b
L.C.M (p,q) = a³ b³
Now, we prove that,
L.C.M (p,q) x H.C.F (p,q) = p q
L.H.S = L.C.M (p,q) x H.C.F (p,q) 
L.H.S = (a³ b³) x (a² b)
L.H.S = a⁵ b⁴
R.H.S = (a² b³) x (a³ b)
R.H.S = a⁵ b⁴
which shows that L.H.S = R.H.S
Hence proved.
Thanks.