First, express

as the sum of

and

.

.

.

From this, I can express

with the smallest positive integer coefficients modulo 10.

ie.

Notes:

So, as an example, if I know that my 2nd row contains elements "2", then I know my entries in my result row are

(It'll be a row of 4's.)

When adding rows, the process can be simpler using coordinate forms instead of writing out r_i's for each row, and if I 'drop the 10's' along the way. For example,

has the same remainder as

modulo 10. I don't show this here, but it was a good verification step to ensure my last row was correct.

To be aware of the cycle that exists, simply set rows

EDIT:// Expressed in a more elementary way that doesn't use modulo but means the same thing:

The last digits of

is what we'd see as a result of the operation described in the problem, for each entry.