Using the digits 1 up to 9, two numbers must be made. The product of these two numbers should be as large as possible. All digits must be used exactly once. Which are the requested two numbers. ?

The digits of the requested two numbers obviously form descending sequences. Furthermore, if you have two pairs of numbers with equal sums, the pair of which the numbers have the smallest absolute difference, is the one of which the numbers have the largest product. Using this knowledge, the two numbers can easily be constructed by placing the digits one by one, starting with 9 and ending with 1:

9

8 -> 96

87 -> 964

875 -> 9642

8753 -> 9642

87531

Conclusion: the requested two numbers are 9642 and 87531 (and the product of these two numbers is 843973902).