$n \times n$ matrix whose entries $\in \{1,2\}$, such that $7$ divides the sum of every column and $5$ divides the sum of every row
Solution 1:
Suppose such a $9 \times 9$ matrix exists. We consider every column of the matrix. The sum of every column is at least $9$ and at most $18$. Since the sum is divisible by $7$, then the sum must be $14$.
As a result, the sum of the numbers in the entire matrix is $9 \times 14 = 126$. But, as you have mentioned, the sum should have been divisible by $5$ too. So, no such matrix exists.