Magical power of cubes-3

Magical properties of cubes -3

We know that n³ is divisible by n with a quotient n². It is quite interesting to note that n³ is divisible by n+1 with a remainder n and for all n ≥ 3, n³ is divisible by n-1 with a remainder 1. When n ≥ 2 ,n³ is perfectly divisible by 8 for all even n, but when n is odd, it is n³- n that is divisible by 8. It is found that n³- n is also perfectly divisible by 6 for all values of n (n≥ 2). It is found that n³ is divisible by 7 giving remainder 1 for all n in the form (1+7m)or(2+7m) or (4+7m),-1 for (3+7m) or (5+7m) or (6+7m) and 0 for all 7m ,where m = 0,1,2,3,……

Similar mathematical properties are found among squares  some of them are summarized below.

n² is perfectly divisible by n.  When n ≥ 3 ,n² is divisible by n-1 and n+1 with a remainder 1 and divisible by n+2 with a remainder 4. When n ≥ 7 ,n² is divisible by n + 3 with a remainder 9 and in general when n ≥ m² – m + 1, n² is divisible by n+m with a remainder m² .