Collection of prime factorization puzzles

The number below each column is the product of all the numbers in that column. The number to the right of each row is the product of all the numbers in that row. Fill in the grid below with numbers 1-9, using each exactly once.

For this puzzle type, we could also make impossible puzzles, quite easily. For example, 256 can't appear because it would require 8 factors of 2, but 1-9 contains just 7. As another example, the numbers 14 and 21 can't both appear in columns, as 7 is a prime factor of both, but there is only one factor of 7 in 1-9, namely 7 itself.

This puzzle type could be extended to hexagons as well, although I'm not sure whether such puzzles would be easier or harder than those on a square grid.