In a $∆$OAB, E is the midpoint of OB and D is a point on AB such that AD: DB = 2 :1. If OD and AE intersect at P, then OP : PD = l :m with least possible positive integer l,m values. Then value of l.m is 

$\overline{\mathrm{E}}=\frac{1}{2}\overline{\mathrm{b}}\mathrm{\&}\overline{\mathrm{OD}}=\frac{2\overline{\mathrm{b}}+\overline{\mathrm{a}}}{3}$
$\overline{\mathrm{OP}}=\mathrm{\lambda }\overline{\mathrm{PD}}=\mathrm{\lambda }(\overline{\mathrm{OD}}-\overline{\mathrm{OP}})\Rightarrow \overline{\mathrm{OP}}=\frac{\mathrm{\lambda }}{\mathrm{\lambda }+1}\left(\frac{2\overline{\mathrm{b}}+\overline{\mathrm{a}}}{3}\right)......\left(1\right)$

$\overline{\mathrm{AP}}=\mathrm{\mu }\overline{\mathrm{PE}}\Rightarrow \overline{\mathrm{OP}}-\overline{\mathrm{OA}}=\mathrm{\mu }(\overline{\mathrm{OE}}-\overline{\mathrm{OP}})\Rightarrow \overline{\mathrm{OP}}=\frac{2\mathrm{a}+\mathrm{\mu b}}{2(\mathrm{\mu }+1)}$
From (1) & (2)$\frac{\mathrm{\lambda }(\overline{\mathrm{a}}+2\overline{\mathrm{b}})}{3(\mathrm{\lambda }+1)}=\frac{2\overline{\mathrm{a}}+\mathrm{\mu }\overline{\mathrm{b}}}{2(\mathrm{\mu }+1)}$
Comparing coeff of $\overline{\mathrm{a}}\text{ and }\overline{\mathrm{b}}\Rightarrow \frac{\mathrm{\lambda }}{3(\mathrm{\lambda }+1)}=\frac{1}{\mathrm{\mu }+1}$........(3)
$\frac{2\mathrm{\lambda }}{3(\mathrm{\lambda }+1)}=\frac{\mathrm{\mu }}{2(\mathrm{\mu }+1)}$      ....(4)
Solving (3) and (4)
$\mathrm{\mu }=4;\mathrm{\lambda }=\frac{3}{2}\therefore \mathrm{\lambda }:1=3:2$