A question asking to prove an array is not a Latin square

Question

I am trying exercises of Ch. 10 of Combinatorics by Richard Brualdi, and I am struck on this question.

Let $n$ be a positive integer and let $r$ be a nonzero integer in $Z_n$ such that $\gcd(r, n)\ne 1$ . Show that array $A$ consructed as $a_{i,j} = r \times i+j \pmod n$ is not a Latin square .

I am thinking to show that two values in some row or column are same, which will prove that it's not a Latin square. So, I choose $a_{i, j} = r\times i + j +kn$ and trying to prove $it = r× i+ m + k'n Or = r×l + j + k"n$ but I am unable to implement this approach. Any help would be greatly appreciated.

Also, if someone has another method to solve that too is equally good.

Answer 1

Here's what we get for $n = 6, r = 2$.

The reasonable guess is that a column would get repeats, hence isn't a Latin square.

Can you prove that?

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Let $\gcd (r,n) = d > 1$, so $ r = da, n = db$.
Then, $ r_{i,j} \equiv r_{i+b,j } \pmod{n}.$