Solution 07807a88-d5cb-430e-be3f-0c93f18f6d2e

Points: 2 pts

Lemma 1: $9(x-1)=14(y-1)$.

At row $x$ and column $y$, the two numbers will be
\[
\bigl(N(x-1)+y,\;M(y-1)+x\bigr).
\]
Setting them equal and simplifying, we find
\[
(N-1)(x-1)=(M-1)(y-1).
\]
For a $141\times 91$ grid, we get
\[
90(x-1)=140(y-1)
\quad\Longrightarrow\quad
9(x-1)=14(y-1),
\]
which is the desired result.

Lemma 2: $(x,y)=(14k+1,\,9k+1)$ for some integer $k$.

Since $9$ and $14$ are coprime, we can parametrize $x$ and $y$ as
\[
(x,y)=(14k+1,\;9k+1),\qquad k\in\mathbb{Z}.
\]
We know the bounds on $x$ and $y$ are
\[
\begin{aligned}
1&\leq x\leq 141,\\
1&\leq y\leq 91.
\end{aligned}
\]
Therefore, $k$ can be
\[
k\in\{0,1,\ldots,10\}.
\]
Clearly, there are $11$ values of $k$, each giving one matching square.

Final answer:
11