Can we form a rectangle with integral lengths using an odd number of copies of this domino?

Question

Question:

This figure is made up of 6 unit cells. Can we form a rectangle with integral lengths using an odd number of copies of this domino? Rotating and flipping of the figure is allowed.

This question is a past year question from a school competition.

After trying for a while, one can hypothesize that the answer is no. However, I have no idea how one would go and prove this result. I tried all kinds of coloring like chessboard coloring and alternate row/column coloring. I also tried many numberings but to no avail.

Besides coloring and numbering, I do not see any other approach. Any help on this problem would be appreciated.

Answer 1

You can use $11$ copies of that hexomino to make a $6\times 11$ rectangle:

This seems to be the smallest solution. A $5\times 18$ rectangle can be seen in Jean Marie's answer, and here is a $9\times 10$ solution:

I found these solutions using my Polyform Puzzle Solver software.

Answer 2

Here is a solution in a $5 \times 18$ rectangle involving 15 copies of your "polyomino" :

I knew that such a solution exist according to the results gathered at the end of this article but it took me some time before I find it... In this article, the existence of a solution in a $6 \times 11$ rectangle (explicitly found by Japp Sherphuis) is mentionned and, as well, the existence of a solution in a $9 \times 10$ rectangle which remains to be found...