4. Determine the natural numbers $a$ and $b$ for which $a<b$, $ab=13824$, and $\operatorname{GCD}(a, b)=24$.
Solution. From $\operatorname{GCD}(a, b)=24$ it follows that $a=24 x$ and $b=24 y$, where $\operatorname{GCD}(x, y)=1$. Furthermore, substituting into $a b=13824$ we get $24 x \cdot 24 y=13824$, i.e., $x y=24$, where $\operatorname{GCD}(x, y)=1$. But, $24=2 \cdot 2 \cdot 2 \cdot 3$ and since $a<b$ we have $x<y$, so from $\operatorname{GCD}(x, y)=1$ and $x y=24$ it follows that $x=1$, 
$y=24$ or $x=3, y=8$. Finally, the required numbers are $a=24, b=576$ or $a=72, b=192$.