prove Orthogonal Latin Squares

Question

Suppose that $n$ is an odd positive integer with $n \geq 3$. Let $A$ be the $n \times n$ Latin square whose rows and columns are indexed by the elements of $\mathbb Z_n = \{0, 1, 2, \ldots, n - 1\}$ with $(i, j)$-entry $a_{i, j} = i + j$ and let $B$ be the $n \times n$ array with $(i, j)$-entry $b_{i, j} = i + 2j$ (where the addition and multiplication is that of $\mathbb Z_n$). Show that $B$ is a Latin square and the Latin squares $A$ and $B$ are orthogonal.

I'm not too sure how to go about answering this question, however, I think I should prove it by induction??

Answer 1

If the Latin squares were not orthogonal then $(a_{i,j},b_{i,j})=(a_{r,c},b_{r,c})$ for some $i,j,r,c$ with $(i,j) \neq (r,c)$.

So the question is essentially asking you to show:

If $a_{i,j}=a_{r,c}$ and $b_{i,j}=b_{r,c}$ for some $i,j,r,c$, then $i=r$ and $j=c$.

Or equivalently, to show:

If $i+j=r+c$ and $i+2j=r+2c$ for some $i,j,r,c$, then $i=r$ and $j=c$.

Here you can forget about Latin squares, and just do the arithmetic.