# Michael's divisible grids

Probability Level 4

There are $N$ ways to fill each square of a $2 \times 3$ rectangular grid with positive integers from 1 to 12, such that the sum of integers in each row is a multiple of 2 and the sum of integers in each column is a multiple of 3. What are the last three digits of $N$?

This problem is posed by Michael T.

Details and assumptions

The integers need not be distinct. As an explicit example,

$\begin{array}{ | l | l | l | } \hline 12 &12 &12 \\ \hline 12 & 12 & 12\\ \hline \end{array}$

is a valid solution.

A $2 \times 3$ grid has 2 rows and 3 columns.

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