mathematics - Primes, composites, and an 11-by-11 square

I really enjoyed attempting to find a solution to Filling an 11-by-11 square (and am a fan of Gamow's square puzzles), so I was inspired to make a puzzle in the same spirit.

Is it possible to fill all $121$ entries of an $11\times11$ grid with the numbers $0$, $1$, and any composite number less than or equal to $10$ such that the row sums and column sums contain the first $22$ prime numbers?

Answer

If the row sums and the columns sums contain the first 22 prime numbers, their sum must be equal to the sum of the first 22 prime numbers, which is odd.

However, the sum of the row sums is equal to the sum of the column sums, because they are both equal to the total of all the numbers in the grid. Therefore, their sum must be even.

Therefore, the row and column sums cannot be the first 22 prime numbers.