I found a solution trying to minimize prime factors. And finding a balance between the minimum value and minimum number of factors.
I noticed that some fields are connected, in that multiplying one of them results in the multiplication of fixed other fields. There are three such patterns:
-C, D, H, L, and M (and any mirror image of that)
-A, E, G, L, M (...
Edit: In an effort to find the minumum, here is a much smaller solution in which the mutual product is
As MKBakker pointed out we could further reduce this by dividing each of the entries 4,8,16,96 and 192 by 2 to get a mutual product of
although they have subsequently improved on this.
And here is the number you are probably thinking of:
It works only for $ab$ where $a \le b$. I suppose that it is a mistake in the problem statement. Others have proven that as it is, the problem is unsolvable.
And here is how I came to that number.
PS: I have been playing with this problem. You can extend it to $ab$ with $a > b$ with the ...