Question

If two positive integers p and q are written as p=^2^3 and q=^3^2 and a and b are two prime numbers, then verify that LCM(a,b)X HCF(a,b)=pq.

Answer 1

Answer:

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.