(a) Find the smallest number which is divisible by both 644 and 462.

\[\begin{aligned} & 462=2 \times 3 \times 7 \times 11 \\ & 644=2^{2} \times 7 \times 23 \\ & \operatorname{LCM}(462,644)=2^{2} \times 3 \times 7 \times 11 \times 23=21252 \end{aligned} \]

\(\displaystyle \therefore\) Smallest number which is divisible by both $\displaystyle 462$ and $\displaystyle 644$ is $\displaystyle 21252$

Let the two numbers be 4x and 5x where x is common factor

Now HCF = $\displaystyle 11$

\[\therefore \mathrm{x}=11 \]

Numbers are $\displaystyle 44$ and $\displaystyle 55$

\[\operatorname{LCM}(44,55)=\frac{44 \times 55}{11}=220 \]