ALL ABOUT CUBE NUMBERS

Creative thoughts-4 (14 th July 2010)

1.A thought to think

The beauty of any outer space is the embeded objects within it while the beauty of any inner space is the embeded thoughts.

This is the information broadcast non-stop by our universe. It means that our universe and its structure both macro and micro are eternal.

2.Recreational mathematics

Any number (n) multiplied twice by itself gives a cube number (n³). n³ = n x n x n

1,8,27,64,125,216,343,512,729........ are the first few cube numbers.

The cube numbers have many fascinating mathematical properties. Some of them are illustrated here.Unlike the square numbers, the cube numbers can end with any digit from 1 to 9. However, its digital root can take only any one of the three values 1,8, 9. ( Digital root of a number is the sum of all the digits of the number reduced to single digit by similar repeated operations ).

4³ = 64 ; 6+ 4 = 10; 1+ 0 = 1 5³ = 125 ; 1+2+5 = 8 6³ = 216 ; 2+1+6 = 9

I f a cube number ends with 1,4,5,6, or 9 , its cube root will also be ending with the same number.

11³ = 1331 ; 21³ = 9261 ; 31³ = 29791 14³ = 2744 ; 24³ = 13824 ; 34³ = 39304 15³ = 3375 ; 25³ = 15625 ; 25³ = 42875 16³ = 4096 ; 26³ = l1576 ; 36³ = 46656 19³ = 6859 ; 29³ = 24389; 39³ = 59319

Cube numbers can be obtained by multiplying any three successive numbers in natural series and adding the middle number with the product.

0 x1 x 2 + 1 = 1 = 1³ 1 x 2 x 3 + 2 = 8 = 2³ 2 x 3 x 4 + 3 = 27 = 3³ 3 x 4 x 5 + 4 = 64 = 4³

Generalizing this property we get , ( n -1)n (n+ l) + n = n³

A cube number (n) can be expressed as the sum of n successive odd numbers, which begin from [n² -(n-1)] and end with [ n² + (n-1].

1³ = 1 2³ = 3+5 = (2² -l) + (2² +l) 3³ = 7 +9 + 11 = (3² -2) + 3² + (3² + 2) 4³ = 13 + 15 + 17 + 19 = (4² -3) + (4² -1) +(4²+ 1) + (4² +3)

It can also be shown as a sum of n successive even numbers along with its root, where the series of even numbers begins with (n-1)n and ends with (n-1)(n+2).

1³ = (0) + 1 2³ = (2+4) + 2 3³ = (6+8+10) + 3 4³ = (12+14+16+18) + 4

It is found that all the cube numbers take any one of the forms either 9x ± 1 or 9 x. If the cube root n = (1+3m), where m is any integer, then its cube will always be in the form 9x +1,if n = (2+3m),the cube follows 9x-l. If n = 3m, the cube number will be perfectly divisible by 9.

n³ = 9x + l ; n = (l+3m) n³ = 9x – 1 ; n = (2+3m) } m = 0,1,2,3,4..... n³ = 9x ; n = 3m

All cube numbers can be denoted as the sum or difference of a multiple of ten and that number

2³ = 8 = 1(10) -2 3³ = 27 = 3(l1) -3 4³ = 64 = 6(10) + 4 5³ = 125 = 12(10) + 5 6³= 216 = 21(10) + 6 7³ =343 = 35(10) -7 8³ =512 = 52(10) -8 9³ = 729 = 72(10) + 9

We can delineate a cube number as the sum of a multiples of 6 and its root number, which predicts that n(n²-1) is always divisible by 6.

1³ = 0 x 6 + 1 2³ = 1 x 6 + 2 3³ = 4 x 6 + 3 4³ = l0 x 6 + 4 5³ = 20 x 6 + 5 6³ = 35 x 6 + 6

The multiplier of 6 for a root number n is the sum of the first n triangular numbers. The sum of numbers from 1 in natural series is called n th triangular number and is denoted by Tn .First few triangular numbers are 0,1,3,6,10,15,21,28..... In terms of T a cube number can be pressed out as

n³ = 6Σ Tn + n

If n is odd, the multiplier of 6 is always even. When n is singly even or odd multiples of 2,the multiplier of 6 is odd, and when n is doubly even i.e., even multiples of 2, the multiplier of 6 is even.

The cube numbers show yet another kind of regularity. They are all in the form of 7m ± l or 7m and exhibit it in a cyclic manner.

2³ = 7 x l + 1; 3³ = 7 x 4 – 1

7³ = 7 x49 + 0; 14³ = 7 x 392 + 0

4³ = 7 x 9 + 1; 5³= 7 x 18 – 1

8³ = 7 x 73 + 1 ; 6³ = 7 x 31 - 1

9³ = 7 x 104 + 1 ; 10³ = 7 x 143 – 1

The root numbers whose cube numbers are in the form of 7m ± 1 or 7m are repeating cyclically. The cube numbers of the root numbers (1+7x),(2+7x) and (4+7x) show the form 7m +l and (3 + 7x), (5+7x) and (6+7x) exhibit the form 7m –l where as 7x display the form 7m,where x is any integer.

The cubes of all even numbers (n) are perfectly divisible by 8 and the multiplier is an odd cube number, if n is singly even and an even cube number if n is doubly even.

2³ = 8 (1³) ; 4³ = 8 ( 2³)

6³ = 8 (3³) ; 8³ = 8 ( 4³)

10³ = 8 (5³) ; 12³ = 8 (6³)

If n is odd ,n³ can be expressed as a sum of some multiples of 8 and n.

1³ = 8(0) + l

3³ = 8 (3) + 3

5³ = 8(15) + 5

7³ = 8(42) + 7

9³ = 8(90) + 9

Here the multiplier of 8 for the given root number n is invariably divisible by n.

All odd cube numbers exhibit the following property- a sum of a multiple of 24 and its root number.

For odd n, n³ = 6 m + n = 8 p + n and hence they can be combined together to show that

n³ = 24 (z) + n.

3³ = 24 (1) + 3

5³ = 24 (5) + 5

7³ = 24 (17) + 7

9³ = 24 (30) + 9

11³= 24 (55) +11

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