Question

​​If two positive integer p and q are wriiten as p=a²b³ and q=a³b; a, b are prime numbers, then verify: LCM(p,q) X HCF(p,q)=pq

Answer 1

Dear Student,
Here is the solution of your asked query:
$$\begin{gathered} \mathrm{p}={\mathrm{a}}^2{\mathrm{b}}^3=\mathrm{a}\times \mathrm{a}\times \mathrm{b}\times \mathrm{b}\times \mathrm{b} \\ \mathrm{and}\mathrm{q}={\mathrm{a}}^3\mathrm{b}=\mathrm{a}\times \mathrm{a}\times \mathrm{a}\times \mathrm{b} \\ \mathrm{So},\mathrm{L}.\mathrm{C}.\mathrm{M}.\mathrm{of}\mathrm{p}\mathrm{and}\mathrm{q}=\mathrm{LCM}\left(\mathrm{p},\mathrm{q}\right)=\mathrm{a}\times \mathrm{a}\times \mathrm{a}\times \mathrm{b}\times \mathrm{b}\times \mathrm{b}={\mathrm{a}}^3{\mathrm{b}}^3 \\ \mathrm{And}\mathrm{HCF}\mathrm{of}\mathrm{p}\mathrm{and}\mathrm{q}=\mathrm{HCF}\left(\mathrm{p},\mathrm{q}\right)=\mathrm{a}\times \mathrm{a}\times \mathrm{b}={\mathrm{a}}^2\mathrm{b} \\ \mathrm{Now},\mathrm{LCM}\left(\mathrm{p},\mathrm{q}\right)\times \mathrm{HCF}\left(\mathrm{p},\mathrm{q}\right)={\mathrm{a}}^3{\mathrm{b}}^3\times {\mathrm{a}}^2\mathrm{b}={\mathrm{a}}^{3+2}{\mathrm{b}}^{3+1}={\mathrm{a}}^5{\mathrm{b}}^4 \\ \mathrm{And}\mathrm{p}\times \mathrm{q}={\mathrm{a}}^2{\mathrm{b}}^3\times {\mathrm{a}}^3\mathrm{b}={\mathrm{a}}^{2+3}{\mathrm{b}}^{3+1}={\mathrm{a}}^5{\mathrm{b}}^4 \\ \mathrm{Therefore},\mathrm{LCM}\left(\mathrm{p},\mathrm{q}\right)\times \mathrm{HCF}\left(\mathrm{p},\mathrm{q}\right)=\mathrm{p}\times \mathrm{q} \\ \mathrm{Hence}\mathrm{verified}. \end{gathered}$$
Regards