That should be obvious.
The last non-zero digit will be 2,4,6, or 8.
You can easily generate a sequence and easily find the last two digits that can occur; then continue that to determine the last three non zero digits.
It is not difficult and the exercise is good.
With that information, you can determine the periodic occurence of the last non-zero digit, the last two non-zero digits, the last three non-zero digits, etc...
Then: you know n, n-1, n-2 , n-3, etc.
You can map the last sequence of digits to the last digits of the factorial.
Hope that helps.