Answer :

First, we’ll find HCF (p, q), when p = a^{2}b and q = a^{3}b.

p = a^{2}b^{3}

= a × a × b × b × b

q = a^{3}b

= a × a × a × b

Common factors are a, a and b.

Thus, HCF (p, q) = a × a × b = a^{2}b …(i)

Now, let’s find LCM (p, q).

We have

LCM (p, q) = a × a × a × b × b × b = a^{3}b^{3} …(ii)

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

Taking LHS:

LCM (p, q) × HCF (p, q) = a^{2}b × a^{3}b^{3} [∵ from equations (i) & (ii)]

= a^{2+3}b^{1+3}

= a^{5}b^{4} …(iii)

Taking RHS:

pq = a^{2}b^{3} × a^{3}b

= a^{2+3}b^{3+1}

= a^{5}b^{4} …(iv)

∵ LHS = RHS

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

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