What is the largest possible product for 2 even integer whose sum is 32

The largest possible product is  16 x 16 = 256 .

You may not see it right away, but this is EXACTLY the same
as asking "What is the largest possible area for a rectangle
whose length and width add up to 32 ?".  The answer to that
one is:  A square that's 16ft x 16ft = 256 ft².

31 is prime. and it’s prime factorization is just 31.