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.

 

 

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 .

Magical properties of Cubes-2

We know that a sum of two cubes can be expressed as a product of two  numbers, where one of them is the sum of its root numbers. i.e., a³ + b³ = (a+b) (a² – ab + b² ) . When a = b, the product can be expressed as a palindromic series. It is illustrated with few examples.

 2³ + 2³ =16= 4 + 8 + 4 ( 3 terms)

3³ + 3³ =54= 6 + 12 + 18 + 12 + 6 (5 terms)

4³ + 4³ =128= 8 + 16 + 24 +32 + 24 + 16 + 8 ( 7 terms)

5³ + 5³ =250= 10 + 20 + 30 + 40 + 50 + 40 + 30 + 20 +10 (9 terms) and so on.

It is noted that the palidromic series begins with twice the root number (2a) in the given sum of two identical cubes a³ + a³ = 2a³ and contains 2a-1 terms.

When one of the root numbers is 1, then it shows a kind of arithmetic series. E.g.,

1³ + 2³ =9= 3 + 6 (=2x3)

1³+ 3³ = 28= 4 + 8 + 16 (=4x4)

1³ + 4³ = 65 = 5 + 10 + 20 + 30 (=6x5)

1³ + 5³ = 126 = 6 + 12 +24 + 36 + 48 (=8x6)

1³ + 6³ = 217 = 7 + 14 + 28 + 42 + 56 + 70  (=10x7) and so on.

Magical properties of cube numbers

Magical properties of cube numbers.

There are many hidden properties among cube numbers.Few are described below.

Different ways of expressing n³ 

4³ = 4² x 4 + 4x0²  = 16 x 4 +0

    = [4²  - 1x4](4+1) + 4 x 1² = 12 x 5 + 4 x 1

  =[ 4²- 2x4](4+2) + 4 x 2²  = 8 x 6 + 4 x 4

= [4²- 3 x4](4+3) + 4 x 3²  = 4 x 7 + 4 x 9

For 5³

5³ = 25 x 5 + 0

    = 20 x 6 + 5 x 1

= 15 x 7 + 5 x 4

= 10 x 8 + 5 x 9

= 5 x 9 + 5 x 16

1³ = 1

2³ = 1 +(1+6)= 1+7 = 8

3³= 1 +(1+6) +(1+6+12)= 1+7+19= 27

4³ = 1+(1+6)+(1+6+12)+(1+6+12+18)= 1+7+19+37 =64

5³= 1+(1+6)+(1+6+12)+(1+6+12+18)+(1+6+12+18+24)= 1+7+19+37+61= 125

1³ = 0²(6) +1

2³ = 1²(6)+2

3³= 2²(6) +3

4³ = (1² +3²) 6 + 4

5³ = (2² + 4² ) 6 + 5

6³ = (1² +3² +5²) 6 +6

7³ = (2²+4²+6²) 6 + 7

8³ = (1²+3²+5²+7²) 6 +8

9³ = (2²+4²+6²+ 8² )6 +9

10³ = (1² +3² +5² +7² + 9²) 6 + 10

In general  a³ = [ 1² +3² + 5² + ……. (a-l)²] 6 + a if a is even

                        = [ 2² + 4² + 6² +    (a-1)²] 6 + a if a is odd