Product of two positive integers a and b is 192. Let$g = hcf (a, b)$ and $l = lcm (a, b).$

 If the ratio of A.M. between g and l to the H.M. between g and l is $\frac{169}{48}$then smaller of a, b is 

we have 

$gl=ab=192$

Also, $\frac{1}{2}(g+l):\frac{2gl}{g+l}=169:48$

$\Rightarrow \frac{(g+l{)}^2}{4gl}=\frac{169}{48}$

$\Rightarrow (g+l{)}^2=\frac{169}{48}(4\left)\right(192)={13}^2(4{)}^2$

$\Rightarrow g+l=52$

Note that g, l are roots of  

$t^2-(g+l)t+gl=0$

or $t^2-52t+192=0$ 

$\begin{array}{l}\Rightarrow t=4,48.\\ \therefore g=4,l=48\end{array}$

Let $a=4a_1,b=4b_1,$where $hcf\left(a_1,b_1\right)=1$

Also, $4a_1{b}_1=48\Rightarrow a_1{b}_1=12$

$\Rightarrow a_1=1,b_1=12\text{ or }{a}_1=3,b_1=4$

Thus, smaller of a, b, is 4 or 12.