Coordinate where policeman catches thief if possible A policemen and a thief  are standing on x-axis with Policeman current position as (X1,0) and Thief is standing at (X2,0). Thief always runs away from Policeman . Policeman knows Thief is faster than him but this time Policeman is determined and chases Thief. 
So for given coordinates of Policeman  and thief with their velocities V1 and V2 we need to tell the coordinates at which Policeman catches Thief in integral time and if not possible then also tell that its "not possible".
Example : If X1=4 and X2=20 and V1=10 AND V2=1 then here answer is (14,0)
Explanation :
At time t=0 coordinates of Policeman and Thief : (4,0) and (20,0)
Similarly, At time t=1: (14,0) (21,0) At time t=2: (24,0) (22,0) 
thus answer is :(14,0)
Example 2 : 
If X1=0 and X2=4 and V1=2 AND V2=2 then here answer is NOT POSSIBLE.
 A: If $|x_1-x_2|=0$ answer in Yes. If $|x_1-x_2|>0$ and $v_2<=v_1$ then  answer is No.The last case remains $|x_1-x_2|>0$ and $v_2>v_1$. 
For this case suppose the thief has run distance $d$ when he is caught, so police has moved $d+|x_1-x_2|$.They have attained this in say time $t$ we get
$$v_1t=d$$
$$v_2t=d+|x_1-x_2|$$
we get by both
$$v_2t=v_1t+|x_1-x_2|$$
$$t=\frac{|x_1-x_2|}{v_2-v_1}$$
so if $|x_1-x_2|$ is divisible by $v_2-v_1$ then answer is Yes,otherwise No.
Yes means thief is caught.
Say if thief gets caught the we know $t$, so we know $d$ from first equation.If $x_1<=x_2$ then co-ordinate is $(x_1-d,0)$ otherwise $(x_1+d,0)$.
A: Let $x_1, x_2, v_1, v_2 \in \mathbb{Z}$.
You need to find when $\exists \; t \in \mathbb{Z}$ with $t \geq 0$ such that: 
$
\begin{align*}
& (x_1 + v_1 t) = (x_2 + v_2 t) \\
\Leftrightarrow \qquad & (v_1 - v_2)t = x_2 - x_1
\end{align*}$
This gives 2 cases. Case 1: If $(v_1 - v_2)=0$, then for $t$ to exist we must have that $x_1 = x_2$. In which case the equation is satisfied $\forall t \in \mathbb{Z}$. If $x_1 \neq x_2$, then $\not \exists t$.
Case 2: If $(v_1 - v_2) \neq 0$, then $t = \frac{x_2 - x_1}{v_1 - v_2}$. Then what we need to happen for $t$ to be an integer is clear: $(v_1 - v_2) \mid (x_2 - x_1)$. We also need $t \geq 0$. Since you said that the thief is faster than the policeman $v_1 \leq v_2$.
So for $t \geq 0$ we must have that $x_1 \geq x_2$.
To get the coordinates just plug in the value of $t$. The policeman will be at $(x_1 + v_1 t)$, the thief at $(x_2 + v_2 t)$.
