Magical properties of cubes-4

Magical properties of cubes-4

 

We find some regularities in the divisibility of cube numbers by different divisors, where the remainder has cyclic variation. When n³ is divided by 2, there is no remainder for all even n, and 1 for all odd n.The remainder varies cyclically 1 and 0. When divided by 3, the remainder varies repeatedly in a cycle 1,2,0 for  n = (1+3m),(2+3m) and 3m (where m = 0,1,2,3,4….) respectively. When the divisor is 4, the cyclic variation of the remainder becomes 1,0,3,0 for n = (1+4m),(2+4m),(3+4m) and 4m respectively. With 5 as divisor, the cyclic variation of the remainder is in the form 1,3,2,4,0 for n = (1+5m),(2+5m),(3+5m),(4+5m),5m respectively. Similarly, one can show that the cyclic variation of the remainder is 1,2,3,4,5,0 for the divisor 6; 1,1,6,1,6,6,0 for the divisor 7;1,0,3,0,5,0,7,0 for the divisor 8; 1,8,0 for the divisor 9; and so on.