Let $(a_1,b_1),(a_2,b_2)\cdots (a_{13},b_{13})$ be those points. For the coordinates of the centroid to be integers, the sum of all x coordinates and the sum of all y coordinates both should be divisible by 3. Let us divide the no.s $ a_1, a_2,\cdots a_{13}$ into 3 groups on the basis of modulo 3. So by PHP, there exists 5 of those no.s which are equivalent to each other modulo 3. Now of those 5 no.s taking any three would yield the sum which is a multiple of 3. Now come to the y coordinates. Divide the y coordinates of the points with x coordinates being the 5 selected no.s, into 3 groups on the same basis. Observe that among those 5 y coordinates if any 3 are of same group, then the sum will be divisible by 3, and we would be done. So taking the other case. In this case, there is atleast 1 no. from each group. Taking them together, we find the sum to be divisible by 3. So we are done.