Find the unknown digits A and B of the five-digit number {71A1B}, which is divisible by 45

Find the unknown digits A and B of the five-digit number {71A1B}, which is divisible by 45. Of all such numbers in your answer, select the one with the largest product of the digits.

We are considering the number 71A1B. We need this number to be divisible by 45. So, the last digit can be either 5 or 0.
Consider the options with the last digit 5 ​​(I round the numbers after the decimal point, since they are no longer important here).

71015/45 = 1578.1

71115/45 = 1580.3

71215/45 = 1582.6

71315/45 = 1584.8

71415/45 = 1587

We have already found one number. In this option, you can no longer look for it, since there will no longer be whole options divisible by 45 in this row.
Now let’s consider the options with the last digit 0. There will be 2 such options (71010 and 71910), but they are not of interest to us – after all, in the answer you need to indicate the one with the greatest product of numbers. And since 0 is already present in these variants, their product can only be equal to 0!

Answer: 4 and 5.