Fun with Mathematics
Cube of a number
The cube of a number can be expressed by many ways which have been illustrated in one of the previous articles under this title. Here is yet another property for cubes. It is found that the value of n3 – n [ = (n-1)n(n+1)] is always a multiple of 6, since 6 will always be a factor for the product of any three successive numbers. The multiple behaviour shows another inherent and unique property . This can be mathematically expressed as,
a=0;x=α
n3 - n = ∑ a(6x)
a=n-1;x=1
13 = 0x6 + 1 = 1
23 = 1x6 + 2 = 8
33 = 2x6 + 1x12 + 3 = 27
43 = 3x6 + 2x12 + 1x18 + 4 = 64
53 = 4x6 +3x12 + 2x18 + 1x24 + 5 = 125
63 = 5x6 + 4 x12 + 3x18 + 2x24 + 1 x30 + 6 =216
This can also be expressed in terms of multiples of six where the multiplicand can be expressed as a sum in a systematic way.
13 – 1 = (0) 6
23 – 2 = (1) 6
33 – 3 = (2+2) 6
43 – 4 = (3+4+3)6
53 – 5 = (4+6+6+4)6 and so on.
