power numbers and its divisibility

Power numbers and its divisibility

When a seventh power of a number ‘a’ is divided by 7, the remainder will be equal to that number itself when a < 7 and (a-7n) when a > 6 where n is the  whole number of 7’s in ‘a’ . e.g.,

1⁷ = 1 = 7(0) + 1

2⁷ = 128 = 7(18) + 2

3⁷ = 2187 = 7(312) + 3

4⁷ = 16384 = 7(2340) + 4

5⁷ = 78125 = 7(11160) + 5

6⁷ = 279936 = 7(39990) + 6

When a = 7 or > 7, a= (7n+d)

a⁷ = (7n+d)⁷ and the remainder is determined by d⁷ / 7 and it is equal to d itself.

7⁷ = 823543 = 7 (117648) + 7

                      = 7 (117649) + 0

8⁷ = 2097152 = 7 (299593) + 8

                        = 7 (299594) + 1

 When a fourth power of a number ‘a’ is divided by (a+1) ,the remainder is 1 for all values  of a.

a=1; a⁴ = 1 = 2(0) + 1

a=2; a⁴ = 16 = 3(5) + 1

a=3; a⁴ = 81 = 4(20) + 1

a=4; a⁴ = 256= 5(51) + 1

a=5; a⁴ = 625 = 6(104) + 1

a=6; a⁴ = 1296 = 7(185) + 1

a=7; a⁴ = 2401 = 8(300) + 1

a=8; a⁴ = 4096 = 9(455) +1

The quotient is equal to a³ – a² + a - 1