If two positive integers p and q are written as p = a²b³ and q = a³b; a, b are prime numbers, then verify:

LCM (p, q) × HCF (p, q) = pq

First, we’ll find HCF (p, q), when p = a²b and q = a³b.

p = a²b³ = a × a × b × b × b

q = a³b = a × a × a × b

Common factors are a, a and b.

Thus, HCF (p, q) = a × a × b = a²b …(i)

Now, let’s find LCM (p, q).

We have

LCM (p, q) = a × a × a × b × b × b = a³b³ …(ii)

To verify LCM (p, q) × HCF (p, q) = pq,

Taking LHS:

LCM (p, q) × HCF (p, q) = a²b × a³b³ [∵ from equations (i) & (ii)]

= a²⁺³b¹⁺³

= a⁵b⁴ …(iii)

Taking RHS:

pq = a²b³ × a³b

= a²⁺³b³⁺¹

= a⁵b⁴ …(iv)

∵ LHS = RHS

This means that it is verified LCM (p, q) × HCF (p, q) = pq