Let

the minimum such

. I came up with the following upper bound:

with the following solution to the puzzle

If we inspect this table, we see that it satisfies the conditions for divisibility and the upper bound is correct. Now, if a prime

appears explicitly in the table, its neighbors must be divisible by that prime. If the prime appears in the interior, its neighbors must be at least the sizes

. If it appears on an edge, the neighbors are at least

. If it appears in a corner the neighbors are at least

. Using the lowest of these upper bounds

, in an optimal solution we must have that

, or

. That is, the only explicit primes in this table are at most

and

. No other primes can appear explicitly in the table. As we can see, up to 26, the only numbers that do not appear in this table are

and

. Since these numbers are not allowed in an optimal solution, the solution I provided must be optimal. Therefore

.