divisibility of cube numbers (cont.,)

Divisibility of cube numbers (cont.,)

When a³  is divided by 3, the remainder will not only be same as the remainder for a but also it is periodically varying as 1,2,0.

When a³ is divided by (a+3)

a=1, 4(4) – 15

a=2,5(4) – 12

a=3, 6(6) - 9

a=4,7(10) - 6

a=5,8(16) - 3

a=6,9(24) + 0

a=7,10(34) + 3

a=8,11(46) + 6

a=9,12(60) + 9

                                                                            a-2

In general ,the quotient is in the form [4 +  Σ 2n ] and the remainder is 3(a-6)

                                                                                     0        

a-2

 Σ 2n  = (a-1)(a-2)

 0

It shows that a³ = [4+(a-1)(a-2)](a+3) - 3(a-6)