We have:

1x1=1 (the last digit of 1 is 1)

2x2=4 (the last digit of 4 is 4)

4x4=6 (the last digit of 16 is 6)

5x5=5 (the last digit of 25 is 5)

7x7=9 (the last digit of 49 is 9)

So 9x9 = 1 (the last digit of 81 is 1)