Fun with Mathematics

The last digit of number and its  power numbers

The following table gives an idea about the last digit of the first few natural

numbers and its power up to 10

n      n¹      n²      n³    n⁴      n⁵      n⁶      n⁷      n⁸      n⁹   

1      1        1        1      1       1        1        1       1        1

2      2        4        8       6       2        4        8       6        2

3      3        9        7       1       3       9       7        1        3

4      4        6        4       6       4       6       4         6        4

5      5        5        5       5       5       5       5        5         5

6      6        6        6       6       6       6       6        6         6

7      7        9        3       1       7       9       3        1         7

8      8        4        2       6       8       4       2        6         8

9      9        1        9       1       9       1       9        1         9

10    0        0        0       0       0       0       0        0        0

It is noted that

Last digit of  n^x = last digit of  n ^{x +4y}  

Where n, x and y are all any whole numbers It is found that the last digit of a number invariably

exists  at all of its  power  equal to (1+4y).The number ending with 0,1,5 and 6 has same digital ending for all of its power . The numbers ending with 4 and 9 have one digital ending for its odd power and another digital ending for its even power.