Remainder of power numbers on division

Remainder of power numbers on division

The remainder of a number n to the power of p when divided by a divisor d is cyclically varying not only for p but also for n . For example when d = 7 ,the cyclic variation of the remainder is same for n^p  and

(n+ 7x)^p

Remainder of n^p when divided by 7

n/p         0              1              2              3              4              5              6              7              8              9              10

1              1              1              1              1              1              1              1              1              1              1              1

2              1              2              4              1              2              4              1              2              4              1              2             

3              1              3              2              6              4              5              1              3              2              6              4

4              1              4              2              1              4              2              1              4              2              1              4

5              1              5              4              6              2              3              1              5              4              6              2

6              1              6              1              6              1              6              1              6              1              6              1

7              1              0              0              0              0              0              0              0              0              0              0

8              1              1              1              1              1              1              1              1              1              1              1             

9              1              2              4              1              2              4              1              2              4              1              2

10           1              3              2              6              4              5              1              3              2              6              4

It is found that the remainder of (1+7x)^p when divided by 7 is 1,for all values of x and p..The remainder of  n⁶ when divided by 7  is 1 for all values of n except n = 7x. It is 0 for n = 7x