No. Consider the $12$ primes $61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109$. Obviously for any of those $p$, the only $n$ from ${1, 2, ..., 121}$ that is a multiple of $p$ is $n=p$. Since they're $12$, two of them $p, q$ must be in the same row. The product of that row must be equal to the product of some column. But such a column should contain both $p$ and $q$, which are on the same row and thus on different columns.

The answer is negative,The trick is to consider the 12 primes $61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109$ and use PHP.